apply Nat.mul_le_mul
case h
n : ℕ
hn : 2 < n
⊢ (∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) *
      ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
    (2 * n) ^ Nat.sqrt (2 * n) * (4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p)
case h.h₁
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.range (Nat.succ (Nat.sqrt (2 * n))), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
    (2 * n) ^ Nat.sqrt (2 * n)

case h.h₂
n : ℕ
hn : 2 < n
⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤
    4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p