Prove that for every $n \in \mathbb{N}$ and for all real numbers $x_1,x_2....x_n \in \mathbb{R}$ Prove that for every $n \in \mathbb{N}$ and for all real numbers $x_1,x_2....x_n \in \mathbb{R}$,
$$|x_1 + x_2 +....+x_n| \leq |x_1| + |x_2| +...+|x_n|$$
My attempt:
Base case: $n = 1$
$$|x_1| \leq |x_1|$$
Induction step: $n + 1$
$$|x_1 + x_{n+1}| \leq |x_1| + |x_{n+1}|$$
By the triangle inequality we have shown the inequality to be true for $n + 1$
Is this a valid and good proof? Is there something I am missing or should add?

hint
For $ n=1$, it is obvious.
For $ n=2 $, we prove by disjunction of cases that
$$|x_1+x_2|\le |x_1|+|x_2|$$
for the $(n+1)^{\text{th}}$ step, you have to check that
$$|x_1+x_2+...+x_n+x_{n+1}|\le$$
$$|x_1|+|x_2|+...|x_n|+|x_{n+1}|$$
