Show that $\left(1+\frac xk\right)^k$ is monotonically increasing

Note: I don't want to get the full solution, but only a hint.

I have to show that for $x \in [0, \infty)$, the sequence $\left(1+\frac xk\right)^k$ is monotonically increasing. We were already given the hint that we could use the inequality of arithmetic and geometric means, but I don't see how to apply it yet.

I tried to do it by induction. While the case $k = 1$ works easily, I don't know how to start further from here. I already tried to use the definition of the binomial theorem, but it didn't lead me anywhere.

Hint:

Apply AM - GM inequality with $m > k$ to:

$$\left(1+\frac{x}{k}\right)^{k/m} =\left(\frac{k+x}{k}\right)^{k/m} = \left[\left(\frac{k+x}{k}\right)^k \right]^{1/m} \\= \left[\underbrace{\frac{k+x}{k} \ldots \frac{k+x}{k}}_k \underbrace{1 \ldots 1}_{m-k}\right]^{1/m} \\ \leqslant \frac{1}{m}\left(k \frac{k+x}{k } + (m-k)(1) \right)$$