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add lemma for proof 4 #50

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Apr 18, 2024
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add lemma for proof 4 #50

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12 changes: 12 additions & 0 deletions FormalBook/Ch01_Six_proofs_of_the_infinity_of_primes.lean
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Expand Up @@ -229,6 +229,18 @@ theorem infinity_of_primes₄ : Tendsto π atTop atTop := by
-- (2) This implies that it is not bounded
sorry

-- This might be useful for the proof. Rename as you like.
lemma H_P4_1 {k p: ℝ} (hk: k > 0) (hp: p ≥ k + 1): p / (p - 1) ≤ (k + 1) / k := by
have h_k_nonzero: k ≠ 0 := ne_iff_lt_or_gt.mpr (Or.inr hk)
have h_p_pred_pos: p - 1 > 0 := by linarith
have h_p_pred_nonzero: p - 1 ≠ 0 := ne_iff_lt_or_gt.mpr (Or.inr h_p_pred_pos)
have h₁: p / (p - 1) = 1 + 1 / (p - 1) := by
rw [one_add_div h_p_pred_nonzero, sub_add_cancel]
rw [← one_add_div h_k_nonzero, h₁, add_le_add_iff_left,
one_div_le_one_div h_p_pred_pos hk, @le_sub_iff_add_le]
exact hp


/-!
### Fifth proof

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