For $n\in\mathbb N$, let $x_1,\ldots,x_n\in\mathbb R$. Show that...

Question

Show that $$ \sum_{k=1}^n |x^k|\geq |\sum_{k=1}^n x^k| $$

Attempt at proof by induction:

Inductive anchor (n=1)

$ \sum_{k=1}^n |x^k|\geq |\sum_{k=1}^n x^k| $

$|x_1|\geq |x_1| $

Inductive step:

$ \sum_{k=1}^{n+1} |x_k|\geq |\sum_{k=1}^{n+1} x_k| $

By the triangle inequality,

$|\sum_{k=1}^{n} x_k + x_{n+1}|\leq |\sum_{k=1}^{n} x_k| + |x_{n+1}| $

$\leq \sum_{k=1}^{n} |x_k| + |x_{n+1}|$

=$\sum_{k=1}^{n+1} |x_k|$

Answer 1

Your proof is correct but, in order to avoid ambiguities, it is better to write$$\left(\sum_{k=1}^nx_k\right)+x_{n+1}\text{ instead of }\sum_{k=1}^nx_k+x_{n+1}$$and$$\left(\sum_{k=1}^n\lvert x_k\rvert\right)+\lvert x_{n+1}\rvert\text{ instead of }\sum_{k=1}^n\lvert x_k\rvert+\lvert x_{n+1}\rvert.$$