. rw [div_lt_iff (by norm_num)]; norm_cast; linarith
case this
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
⊢ 81 / 2 < ↑n

case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False
case inr
n : ℕ
hn : 0 < n
hn' : 4000 ≤ n
h' : 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
this : 81 / 2 < ↑n
⊢ False