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.