refine' mul_le_mul_of_nonneg_left (lower_monotone_aux n) (by positivity)
n : ℕ
⊢ 4 * (Real.sqrt (↑n + 1 / 4) / Real.sqrt (↑n + 1 + 1 / 4)) ≤ 4 * ((↑n + 1 / 2) / (↑n + 1))

case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4)

case d0
n : ℕ
⊢ 0 < ↑(centralBinom n)

n : ℕ
⊢ 0 < 4
case b0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 4)

case d0
n : ℕ
⊢ 0 < ↑(centralBinom n)

n : ℕ
⊢ 0 < 4