Equality of Integers to the Power of Each Other/Proof 2

Proof
suppose $m > n$.

Write $m = n + x$, where $x$ is an integer.

Then:

From Real Sequence $\paren {1 + \dfrac x n}^n$ is Convergent:
 * $\paren {1 + \dfrac x n}^n$ is increasing and has limit $e^x$.

Hence:
 * $n^x < e^x$

This forces $n = 2$.

We have:
 * $m^2 = 2^m$

showing that $m$ is a power of $2$.

Write $m = 2^k$.

Then:
 * $2^{2 k} = 2^{2^k}$

giving:
 * $k = 2^{k - 1}$

By Bernoulli's Inequality:
 * $2^{k - 1} \ge 1 + k - 1 = k$

where equality holds $k - 1 = 0$ or $k - 1 = 1$.

We can skip Bernoulli's Inequality by induction on $k$ for $k > 2$.

Either way, this gives:
 * $k = 1$ or $2$
 * $m = 2$ or $4$.

We reject $m = 2$ since $n = 2$.

Hence $2^4 = 4^2$ is the only solution.