rw [div_le_iff, mul_assoc, mul_comm _ (_ * _), ← div_le_div_iff, mul_comm, mul_div_assoc,
  centralBinom_ratio, Nat.cast_add_one]
n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 ≤ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3)
n : ℕ
⊢ 4 * ((↑n + 1 / 2) / (↑n + 1)) ≤ 4 * (Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3))

case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)

case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)

n : ℕ
⊢ 0 < 4