proof verification - Prove by mathematical induction $4^n > n+1$

Prove the following by mathematical induction:

$4^n > n+1$, for all integers $n ≥ 1$

Step 1:
$n=1$:

LHS $= 4^{(1)} = 4 $

RHS $= (1) + 1 = 2$

LHS > RHS. ∴ $P(1)$ is true.

Step 2:

Assume $P(k)$ is true for some $k ≥ 1$

$P(k)$: $4^k > k+1$

Step 3: We must show $P(k+1)$ is true.

$n = k+1$: $4^{k+1} > (k+1)+1$

RHS = $(k+1)+1 < 4^k +1 < 4^k + 4^k = 2*4^k < 4*4^k = 4^{k+1}$ = LHS

Hence, $P(k+1)$ is true. Therefore, By Math. Induction $P(n)$ is true for all $n ≥ 1$

Can anyone check if my method is correct or there is a better way to do it. Thank you.