apply mul_le_mul
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3)
case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))

case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3)

case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)

case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))