For which $p \in (0, +\infty)$ is the set $\{x \in \mathbb{R}^n : (\sum_{i=1}^n |x_i|^p)^{1/p} \leq 1\}$ convex?

Question

For which $p \in (0, +\infty)$ is the set $\{x \in \mathbb{R}^n : (\sum_{i=1}^n |x_i|^p)^{1/p} \leq 1\}$ convex?

I should be able to answer this question just using the definition of a convex set:

A set $S$ is said to be convex if for every $x$ and $y$ in $S$, and every $\lambda \in [0,1]$, we have that $\lambda x + (1 - \lambda) y \in S$.

That was easy to see for $p=1$, but I have no clue how to prove it in a more general way.