[{"type":"thinking","thinking":"Concerns:\n1. `Real.tendsto_harmonic_sub_log` — its exact statement: `Tendsto (fun n : ℕ ↦ harmonic n - log n) atTop (𝓝 eulerMascheroniConstant)` — with `Real.log n` — from EulerMascheroni.lean:150. The `harmonic n` — in ℚ or ℝ? The lemma says `harmonic n - log n` — if harmonic n : ℚ then there's a cast. The tendsto is in ℝ so there's a coercion `↑(harmonic n)`. My h2 manipulation: `(fun n => Real.log n - (harmonic n : ℝ)) = fun n => -((harmonic n : ℝ) - Real.log n)` — with the cast ✓.\n2. `harmonic` def: `harmonic n := ∑ i ∈ Finset.range n, (i + 1)⁻¹` — is it ℚ or ℝ?? In Mathlib, `harmonic (n : ℕ) : ℚ := ∑ i ∈ Finset.range n, (i + 1)⁻¹` — I believe it's RATIONAL (in Mathlib/NumberTheory/Harmonic/Defs.lean or Basic — `harmonic` is defined in ℚ). So (harmonic n : ℝ) casts. My hh : (harmonic n : ℝ) = ∑ i ∈ range n, 1/((i:ℝ)+1) — via Nat.cast_sum + the cast of (i+1)⁻¹: Nat.cast_inv, Nat.cast_add, Nat.cast_one — but harmonic is a ℚ sum cast to ℝ: (↑(∑ i, (i+1)⁻¹) : ℝ) — the cast ℚ→ℝ is `Rat.cast_sum`! not Nat.cast_sum! Hmm: (↑(harmonic n) : ℝ) where harmonic n : ℚ — the cast is Rat.cast — `Rat.cast_sum` : (↑(∑ i, f i) : ℝ) = ∑ i, ↑(f i) ✓ — then each term ↑((i+1)⁻¹ : ℚ) = ((i+1 : ℝ))⁻¹ — Rat.cast_inv + Rat.cast_add + Rat.cast_one... — the cast path ℚ→ℝ of i+1 (which is ℚ's OfNat (i+1)? — (i + 1 : ℚ) with i : ℕ cast to ℚ first: ((i : ℚ) + 1)⁻¹ — the harmonic def's (i + 1)⁻¹ has i : ℕ cast into ℚ: (↑i + 1 : ℚ)⁻¹. So the cast chain: ↑((↑i + 1 : ℚ)⁻¹) = (↑(↑i + 1))⁻¹ [Rat.cast_inv] = ... → 1/((i:ℝ)+1) — via Rat.cast_inv, Rat.cast_add? Rat.cast_add : ↑(a + b) = ↑a + ↑b ✓, Rat.cast_one, Rat.cast_natCast (↓(↑i))... Let me use a simpler normalizer: \n```lean\n  have hh : (harmonic n : ℝ) = ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) := by\n    rw [harmonic]\n    rw [Rat.cast_sum]  -- hmm the cast of a sum\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Rat.cast_inv, Rat.cast_add, Rat.cast_one, Rat.cast_natCast]\n    field_simp  -- 1/((i+1):ℝ) form vs (↑(i+1))⁻¹...\n```\nHmm — the harmonic's terms: (∑ i ∈ range n, (i + 1 : ℚ)⁻¹ : ℚ)? — let me check the harmonic def exactly: Mathlib/NumberTheory/Harmonic/Defs.lean: `def harmonic (n : ℕ) : ℚ := ∑ i ∈ Finset.range n, (↑(i + 1))⁻¹`? or `(i + 1)⁻¹` with i : ℕ cast? — the ↑(i+1) would be ℕ→ℚ cast. So the term: (↑(i+1) : ℚ)⁻¹. Cast to ℝ: ↑((↑(i+1):ℚ)⁻¹) — Rat.cast_inv → (↑(↑(i+1)))⁻¹ — hmm — the double cast ℕ→ℚ→ℝ: `Rat.cast_natCast : ((↑n : ℚ) : ℝ) = ↑n` ✓ simp lemma. So:\n```lean\n    rw [harmonic, Nat.cast_sum]  -- NO — the sum is in ℚ; cast to ℝ needs Rat.cast_sum\n```\nHmm — `Nat.cast_sum` is for Nat.cast into a semiring — the sum here is in ℚ then cast to ℝ — it's `Rat.cast_sum (s : Finset) (f) : (↑(∑ i ∈ s, f i) : ℝ) = ∑ i ∈ s, ↑(f i)` ✓. So:\n```lean\n  have hh : (harmonic n : ℝ) = ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) := by\n    rw [harmonic, Rat.cast_sum]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Rat.cast_inv, Rat.cast_natCast, Nat.cast_add, Nat.cast_one]  -- hmm\n    field_simp? no — the forms: LHS (↑((↑(i+1):ℚ)⁻¹):ℝ) → after Rat.cast_inv: (↑(↑(i+1):ℚ):ℝ)⁻¹ → Rat.cast_natCast: (↑(i+1 : ℕ):ℝ)⁻¹ — wait the inner is ↑(i+1) : ℚ — Rat.cast_natCast : (↑(↑n : ℚ))... hmm the lemma names: `Rat.cast_natCast : ((n : ℚ) : ℝ) = n`? hmm — Rat.cast_natCast (n : ℕ) : (↑(↑n : ℚ) : ℝ) = ↑n — hmm the nesting: ↑(i+1) in ℚ is Nat.cast (i+1) : ℚ — then cast to ℝ. Rat.cast_natCast rewrites (↑(Nat.cast k : ℚ) : ℝ) → Nat.cast k : ℝ? I think `Rat.cast_natCast (n : ℕ) [Ring R] : ((n : ℚ) : R) = n`? — hmm — it's `Rat.cast_natCast (n : ℕ) : ((n : ℚ) : α) = ↑n`? — for α := ℝ ✓. Then (↑(i+1):ℝ)⁻¹ → Nat.cast_add + Nat.cast_one → (↑i + 1)⁻¹ — and the target 1/((i:ℝ)+1) — 1/z vs z⁻¹: one_div. Order: rw [Rat.cast_inv, one_div] — let me just use simp + field_simp:\n    simp [Rat.cast_inv, Rat.cast_natCast, Nat.cast_add, Nat.cast_one, one_div]\n```\nThis should normalize both sides to (↑i + 1)⁻¹ forms. Alternatively use `push_cast` — the whole thing is a cast-commutation: \n```lean\n    push_cast\n    ring\n```\nper-term: (↑((↑(i+1):ℚ)⁻¹) : ℝ) = 1/((i:ℝ)+1) — push_cast normalizes the LHS to ((↑i + 1 : ℝ))⁻¹ — and RHS 1/(↑i+1) — then `ring`? — 1/z vs z⁻¹ — push_cast gives (↑i+1)⁻¹ and 1/(↑i+1) — `one_div` — hmm ring knows 1/z = z⁻¹? — ring handles division in fields ✓ (it treats / as ·⁻¹) — so push_cast then ring should close ✓. I'll write push_cast; ring and if that fails adjust. Actually safest: `intro i hi; rw [one_div]; push_cast` — hmm. Let me just write `push_cast; ring` and compile.\n\nHmm — wait, does the harmonic def use (↑(i+1))⁻¹ or (i+1)⁻¹? If it's `(i + 1)⁻¹` with i : ℕ cast to ℚ via OfNat?? — no — i : ℕ, and harmonic's sum is over ℚ — the term (i + 1)⁻¹ needs i cast to ℚ: Nat.cast i — so (↑i + 1 : ℚ)⁻¹. Then cast to ℝ: push_cast handles ✓.\n\n3. dpsiSeqR_eq's h2: Finset.sum_range_succ' : ∑ k ∈ range (n+1), f k = (∑ k ∈ range n, f (k+1)) + f 0 — gives the peeled form with f 0 at the END. My target: Σ_{range(n+1)} 1/(x+m) = 1/x + Σ_{range n} 1/(x+i+1). After rw [Finset.sum_range_succ']: the sum becomes (Σ_{i∈range n} 1/(x+(i+1))) + 1/(x+0)... wait f m := 1/(x + ↑m): f (i+1) = 1/(x + ↑(i+1)) = 1/(x + (↑i+1)) and f 0 = 1/(x + 0) = 1/x. So after the rw: goal LHS-position has (Σ_{i∈range n} 1/(x+↑(i+1))) + 1/(x+↑0) — my e-congruence: 1/(x+↑(i+1)) = 1/(x+i+1) via push_cast; ring_nf — hmm e : ∀ i, 1/(x + (i+1:ℝ)) = 1/(x + i + 1 : ℝ) — push_cast turns ↑(i+1) into ↑i+1 ✓ then ring_nf closes the reciprocal equality? 1/(x + (↑i+1)) = 1/(x+↑i+1) — after push_cast the LHS denominator x + (↑i + 1) — the RHS x + ↑i + 1 — same up to association — ring_nf handles division equality? ring_nf can prove a/b = a'/b' if numerators/denominators match after normalization ✓ should. Then the leftover + f 0 = 1/(x + ↑0) = 1/x — hmm — my structure:\n```lean\n    rw [Finset.sum_range_succ']\n    have e : ∀ i : ℕ, 1 / (x + (i + 1 : ℝ)) = 1 / (x + i + 1 : ℝ) := fun i => by push_cast; ring_nf\n    rw [Finset.sum_congr rfl (fun i _ => e i)]\n    ring\n```\nAfter the first two: the goal is `(Σ_{i∈range n} 1/(x+↑i+1)) + 1/(x + ↑0) = 1/x + Σ_{i∈range n} 1/(x+↑i+1)` — hmm wait — what's the actual goal? The goal is the h2 statement: Σ_{range(n+1)} 1/(x+↑m) = 1/x + Σ_{range n} 1/(x+i+1). After rw [Finset.sum_range_succ']: LHS → (Σ_{i∈range n} 1/(x+↑(i+1))) + 1/(x+↑0). After rw [Finset.sum_congr rfl e]: the sum's terms → 1/(x+↑i+1)... wait — the goal then: (Σ_{i∈range n} 1/(x+↑i+1)) + 1/(x+↑0) = 1/x + Σ_{i∈range n} 1/(x+↑i+1) — hmm the LHS's first sum now matches the RHS's sum? The RHS sum is `∑ i ∈ Finset.range n, 1 / (x + i + 1)` — with `i + 1` as (i:ℝ)+1 via cast — and my e's RHS is `1 / (x + i + 1 : ℝ)` — written as (x + i + 1 : ℝ) — hmm the ascription — e's statement: 1 / (x + (i + 1 : ℝ)) = 1 / (x + i + 1 : ℝ) — the second is x + i + 1 elaborated with i : ℕ cast — (x + ↑i + 1 : ℝ) ✓ same as the goal's `1 / (x + i + 1)` = 1/(x + ↑i + 1) ✓. Then ring closes: (S + 1/(x+↑0)) = (1/x + S): needs 1/(x + ↑0) = 1/x: ↑(0:ℕ) = 0: ring normalizes casts? ring doesn't do casts — the term 1/(x + ↑0) — ↑0 = 0 via Nat.cast_zero — ring might not know. Safer end: \n```lean\n    rw [Finset.sum_congr rfl (fun i _ => e i)]\n    have e0 : 1 / (x + ((0 : ℕ) : ℝ)) = 1 / x := by simp\n    rw [e0]\n    ring\n```\nHmm — the f 0 term: 1/(x + ↑(0)) — with ↑(0:ℕ) — simp [Nat.cast_zero, add_zero] → 1/x ✓.\n\nActually — wait, where does f 0 sit after sum_range_succ'? The lemma: ∑ k ∈ range (n+1), f k = (∑ k ∈ range n, f (k+1)) + f 0 — so the goal becomes (Σ ...) + f 0 = 1/x + Σ ... — f 0 = 1/(x + ↑(0+...)) hmm f is the lambda fun m => 1/(x + ↑m) — f 0 = 1/(x + ↑0) ✓ (beta — does Lean beta-reduce automatically in the goal? after rw the term is `(fun m => 1/(x + ↑m)) 0` — beta-reduced by simp/display — for the e0 rewrite to match I need the beta-reduced form 1/(x + ↑0)... — the rw [Finset.sum_range_succ'] output has the application unreduced? rw produces beta-reduced? Not always. Use simp only [] for beta... — let me just make e0 flexible: `have e0 : (fun m => 1 / (x + (m : ℝ))) 0 = 1 / x := by simp` — covering the unreduced form ✓.\n\n4. h3's manipulation: I want Σ (1/(i+1) − 1/(x+i+1)) = Σ dpsiTermR x i — and then I rewrote [← Finset.sum_sub_distrib] at h3?? — h3 is the statement I PROVE: Σ_{range n} (1/(↑i+1) − 1/(x+i+1)) = Σ_{range n} dpsiTermR x i — and dpsiTermR x i = 1/(↑i+1) − 1/(x+↑i+1) — so the per-term is rfl-ish modulo the casts: (i:ℝ)+1 vs ↑i+1: dpsiTermR's def: 1 / (m + 1 : ℝ) − 1 / (x + m + 1) — with (m+1:ℝ) being ↑m + 1? I wrote `1 / (m + 1 : ℝ)` — that's ↑m + 1 in ℝ ✓ — and my sum's term `1 / ((i : ℝ) + 1) − 1 / (x + i + 1)` ✓ SAME — so the sum_congr per-term: rfl? — dpsiTermR x i unfolds to 1/(↑i+1) − 1/(x+↑i+1) ✓ — the goal's term after the h2-substitution... — hmm wait — I mislabeled. The full dpsiSeqR_eq goal after rw [dpsiSeqR]: \n   Real.log n − Σ_{range(n+1)} 1/(x+↑m) = (Real.log n − ↑harmonic) + (−1/x + Σ_{range n} dpsiTermR x i).\n   After rw [hh] (harmonic → sum form) and rw [h2] (the big sum → 1/x + Σ 1/(x+i+1)):\n   LHS: log n − (1/x + Σ_{range n} 1/(x+↑i+1)); RHS: (log n − Σ_{range n} 1/(↑i+1)) + (−1/x + Σ_{range n} dpsiTermR x i).\n   I need: Σ_{range n} dpsiTermR x i = Σ_{range n} 1/(↑i+1) − Σ_{range n} 1/(x+↑i+1)-ish... my h3 approach: h3 : Σ_{range n} (1/(↑i+1) − 1/(x+↑i+1)) = Σ_{range n} dpsiTermR x i (sum_congr, per-term rw [dpsiTermR]; ring — per-term goal: 1/(↑i+1) − 1/(x+↑i+1) = 1/(↑i+1) − 1/(x+↑i+1) after unfolding — rfl — hmm the dpsiTermR unfold: rw [dpsiTermR] unfolds the DEF — dpsiTermR is a def — `rw [dpsiTermR]`?? for a def, rw [dpsiTermR] uses the equation lemma dpsiTermR.eq_def — should work — or `show`... The per-term goal: 1/((i:ℝ)+1) − 1/(x+(i:ℝ)+1) = dpsiTermR x i — rw [dpsiTermR] on the RHS → 1/(↑i+1) − 1/(x+↑i+1) — then the two sides: `1 / (↑i + 1) − 1 / (x + ↑i + 1)` vs `1 / (↑i + 1) − 1 / (x + ↑i + 1)` — SAME? the LHS goal term: `1 / ((i : ℝ) + 1) − 1 / (x + i + 1)` — the `x + i + 1` is x + ↑i + 1 ✓ same ✓ so after rw [dpsiTermR] the goal is rfl — `ring` closes or rfl — I wrote `rw [dpsiTermR]; ring` ✓ fine.\n   Then `rw [← Finset.sum_sub_distrib] at h3` — sum_sub_distrib : ∑ (f − g) = ∑ f − ∑ g — ← direction on h3's LHS: turns Σ (1/(↑i+1) − 1/(x+i+1)) into Σ 1/(↑i+1) − Σ 1/(x+↑i+1) — so h3 becomes Σ 1/(↑i+1) − Σ 1/(x+↑i+1) = Σ dpsiTermR ✓ then rw [← h3] in the main goal — replaces Σ 1/(↑i+1) − Σ 1/(x+↑i+1) with Σ dpsiTermR — hmm the main goal after hh/h2 rewrites: log n − (1/x + Σ 1/(x+↑i+1)) = (log n − Σ 1/(↑i+1)) + (−1/x + Σ dpsiTermR) — does the pattern Σ 1/(↑i+1) − Σ 1/(x+↑i+1) occur? NO — the two sums are in different places. The rw [← h3] fails! Restructure the final algebra: the goal is an identity between expressions with sums as atoms: A := Σ_{range n} 1/(↑i+1), B := Σ_{range n} 1/(x+↑i+1), C := Σ_{range n} dpsiTermR — with the relation C = A − B (h3 reversed: Σ dpsiTermR = Σ 1/(↑i+1) − Σ 1/(x+↑i+1)). Goal: log n − (1/x + B) = (log n − A) + (−1/x + C). With C = A − B: RHS = log n − A − 1/x + A − B = log n − 1/x − B = LHS ✓. So: rw [h3] where h3 : Σ dpsiTermR = A − B?? — my h3 currently : Σ (1/(↑i+1) − 1/(x+i+1)) = Σ dpsiTermR — hmm — let me restructure: \n```lean\n  have h3 : ∑ i ∈ Finset.range n, dpsiTermR x i =\n      ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) - ∑ i ∈ Finset.range n, 1 / (x + i + 1) := by\n    rw [← Finset.sum_sub_distrib]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [dpsiTermR]\n  rw [h2, h3]\n  ring\n```\nWait — the goal after rw [hh]: LHS has Σ_{range(n+1)} 1/(x+↑m) and RHS has Σ_{range n} 1/(↑i+1) (from hh) + Σ dpsiTermR. rw [h2]: LHS's big sum → 1/x + Σ_{range n} 1/(x+↑i+1)... wait h2 : Σ_{range(n+1)} 1/(x+↑m) = 1/x + Σ_{range n} 1/(x+i+1) — rewrites the LHS occurrence ✓. rw [h3]: rewrites Σ_{range n} dpsiTermR x i (in RHS) → Σ 1/(↑i+1) − Σ 1/(x+i+1) ✓. Then the goal: log n − (1/x + B) = (log n − A) + (−1/x + (A − B)) — ring with A, B atoms ✓ closes.\n\nHmm — careful: dpsiTermR unfold in h3's per-term: goal `dpsiTermR x i = 1/(↑i+1) − 1/(x+↑i+1)`?? — wait — sum_congr's per-term goal: dpsiTermR x i = (1/(↑i+1) − 1/(x+↑i+1))? — the sum_congr rfl h : Σ f = Σ g needs h : ∀ i ∈ s, f i = g i — with f := dpsiTermR x and g := fun i => 1/(↑i+1) − 1/(x+↑i+1) — the goal: dpsiTermR x i = 1/(↑i+1) − 1/(x+↑i+1) — rw [dpsiTermR] unfolds the def → 1/(↑i+1) − 1/(x+↑i+1) = ... rfl closed by rw ✓ (or not — then add nothing; if rw leaves rfl-goal it auto-closes ✓).\n\nHmm — actually rw [dpsiTermR] — dpsiTermR is a def `dpsiTermR (x : ℝ) (m : ℕ) : ℝ := 1 / (m + 1 : ℝ) - 1 / (x + m + 1)` — the equation lemma dpsiTermR.eq_... — rw [dpsiTermR] rewrites dpsiTermR x i → 1/(↑i + 1) − 1/(x + ↑i + 1) — hmm — but the def's (m + 1 : ℝ): I wrote `1 / (m + 1 : ℝ)` — that's ↑m + 1 elaborated IN ℝ? `(m + 1 : ℝ)` with m : ℕ — the + is at ℝ: (↑m + 1 : ℝ) ✓ — so the unfold gives 1/(↑i+1) − ... ✓ matches the goal's `1 / ((i : ℝ) + 1)` ✓.\n\nBUT — one subtlety — the sum_sub_distrib: `Finset.sum_sub_distrib : ∑ i, (f i − g i) = ∑ i, f i − ∑ i, g i` ✓ exists — the ← direction: ∑ f − ∑ g → ∑ (f − g) ✓ my h3's proof: `rw [← Finset.sum_sub_distrib]` on the goal `∑ dpsiTermR = ∑ (1/..) − ∑ (1/..)` — ← direction rewrites the RHS's `∑ .. − ∑ ..` → `∑ (.. − ..)` ✓ then sum_congr ✓.\n\n5. In `hf`: the logGammaSeq unfold — `BohrMollerup.logGammaSeq` — is logGammaSeq in the BohrMollerup NAMESPACE? From the file: `namespace BohrMollerup / def logGammaSeq` ✓ so `BohrMollerup.logGammaSeq` ✓ and `BohrMollerup.tendsto_log_gamma` ✓ (line 293 in the BohrMollerup namespace ✓).\n\nHmm wait — is tendsto_log_gamma in the BohrMollerup namespace? The file: namespace BohrMollerup ... def logGammaSeq ... theorem tendsto_log_gamma at line 293 — need to check if it's inside the namespace — from the sed output at 290-300: `theorem tendsto_log_gamma {x : ℝ} (hx : 0 < x) : Tendsto (logGammaSeq x) atTop (𝓝 <| log (Gamma x))` — with `logGammaSeq` and `Gamma` unqualified in the statement — inside namespace BohrMollerup + namespace Real? The file had `namespace BohrMollerup` at 135-ish. And earlier the file had `namespace Real` too? The structure: `namespace Real` ... `namespace BohrMollerup` (nested) — so full name `Real.BohrMollerup.tendsto_log_gamma`? or BohrMollerup under Real — hmm — and the `Gamma` in the statement is Real.Gamma (inside namespace Real ✓). So full name: `Real.BohrMollerup.tendsto_log_gamma`. With my `open ... Real` (I opened Real in this file ✓ \"open Zeta23 Complex Filter Topology Real\") — so `BohrMollerup.tendsto_log_gamma` resolves ✓ and `BohrMollerup.logGammaSeq` ✓. But careful — with `open Real`, `Real.Gamma` vs `Gamma`... in my file `Real.Gamma` is explicit ✓ fine. And `harmonic` — root or Real? — harmonic is root-level (Mathlib NumberTheory.Harmonic — `harmonic n : ℚ`? — hmm wait — is harmonic ℝ or ℚ? Let me check: Mathlib/NumberTheory/Harmonic/Defs.lean: `def harmonic (n : ℕ) : ℚ := ∑ i ∈ range n, (i + 1)⁻¹`? — and tendsto_harmonic_sub_log is Real.tendsto_harmonic_sub_log ✓ (EulerMascheroni.lean:150 — in namespace Real ✓ with (harmonic n : ℝ) cast).\n\n6. hfg's `(Real.log ∘ Real.Gamma) y` vs tendsto_log_gamma's `log (Gamma y)` — the comp form — defeq ✓ — the hfg goal: Tendsto (fun n => logGammaSeq y n) atTop (𝓝 ((Real.log ∘ Real.Gamma) y)) — (Real.log ∘ Real.Gamma) y = Real.log (Real.Gamma y) — defeq — the term tendsto_log_gamma hy : Tendsto (logGammaSeq y) atTop (𝓝 (Real.log (Real.Gamma y))) — hmm — its statement shows `log (Gamma y)` = Real.log (Real.Gamma y) (inside namespace Real) ✓ — and the lambda fun n => logGammaSeq y n vs logGammaSeq y — eta — defeq ✓ exact should accept; if not, `simpa using`.\n\n7. The hA's TendstoLocallyUniformlyOn for the constant-in-x sequence: I used tendstoLocallyUniformlyOn_iff_forall_isCompact + Metric.tendstoUniformlyOn_iff — the goal after those: ∀ ε > 0, ∃ N, ∀ n ≥ N, ∀ x ∈ K, dist ((fun _ => log n − ↑harmonic n) x) (−γ) < ε — hmm — Metric.tendstoUniformlyOn_iff : TendstoUniformlyOn F f l s ↔ ∀ ε > 0, ∀ᶠ n in l, ∀ x ∈ s, dist (F n x) (f x) < ε — so the goal after rw: ∀ ε>0, ∀ᶠ n in atTop, ∀ x ∈ K, dist ... — my `refine ⟨N, fun n hn x hx => ?_⟩` — the ∀ᶠ form is ∃ N, ∀ n ≥ N, ... — `eventually_atTop` unfolding — the obtain from h2: Metric.tendsto_atTop — h2's form after rw [Metric.tendsto_atTop] at h2: ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (...) (−γ) < ε ✓ obtain ✓ then the goal: ∀ x ∈ K, dist (log n − ↑harmonic n) (−γ) < ε — wait — the ∀ᶠ n in atTop, ∀ x ∈ K, ... — I refine ⟨N, ...⟩ — the ∀ᶠ atTop : ∃ N, ∀ n ≥ N, P n — is that the defeq form? eventually_atTop: (∀ᶠ n in atTop, P n) ↔ ∃ N, ∀ n ≥ N, P n — the anonymous constructor ⟨N, ...⟩ works if the goal unfolds ✓ — and the function application: F n x = (fun _ => log n − harmonic n) x = log n − harmonic n — beta — `simpa using hN n hn` — hN n hn : dist (log n − ↑(harmonic n)) (−γ) < ε — the goal's dist ((fun _ => ...) x) (−γ) — beta — simpa handles ✓ hopefully. Hmm — but wait — h2 after `rw [h2]` (the function-eq rewrite) then `exact h.neg` — h.neg : Tendsto (−(harmonic − log)) → 𝓝 (−γ) ✓ — h : Tendsto (fun n => ↑(harmonic n) − Real.log n) ... — the h2's function is fun n => −(↑harmonic n − log n) — and Tendsto.neg : Tendsto (fun n => −(↑harmonic n − log n)) atTop (𝓝 (−(γ)))? — Tendsto.neg (h : Tendsto f l (𝓝 a)) : Tendsto (−f) l (𝓝 (−a)) — −f = fun n => −f n ✓ — the h.neg's function fun n => −(↑harmonic n − Real.log n) — and h2's stated function fun n => Real.log n − ↑harmonic n — NOT the same — I rewrote via h2 (the funext) — rw [h2] at the GOAL — the goal's function fun n => Real.log n − ↑harmonic n → becomes fun n => −(...) ✓ then exact h.neg ✓ — h.neg : Tendsto (fun n => −(↑harmonic n − Real.log n)) atTop (𝓝 (−γ)) ✓ match ✓.\n\nHmm — but actually the original h := Real.tendsto_harmonic_sub_log — its statement form: Tendsto (fun n : ℕ ↦ harmonic n - log n) ... — with `harmonic n` — is it ↑(harmonic n) or (harmonic n : ℝ)? — the statement writes `harmonic n - log n` — harmonic n : ℚ, log n : ℝ — the subtraction forces a cast ↑(harmonic n) — so the function is fun n => ↑(harmonic n) − Real.log n ✓ my h.neg handling assumed that ✓.\n\n8. hC's `hg.tsum_eqOn ht` — HasSumLocallyUniformlyOn.tsum_eqOn — from derivWithin_tsum's usage: `(hg.tsum_eqOn hb).symm` — HasSumLocallyUniformlyOn.tsum_eqOn : (hg : HasSumLocallyUniformlyOn f g s) → (hb : ? ∈ s) → g ? = ∑' ...? — signature guess: (hb : x ∈ s) : g x = ∑' n, f n x ✓.\n\nOK. Compile and iterate.","signature":""},{"type":"text","text":"Compiling section 2 (interchange) — several fiddly points to iterate on."},{"type":"tool_use","id":"Bash_313","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiDigamma.lean 2>&1 | grep -v \"^$\" | head -50","description":"Compile LiDigamma section 2","timeout":600000},"extra_content":null}]