Why is $\| x\| _1\leq n\| x\| _{\infty}$?

Question

I am trying to show that $\| x\| _1\leq n\| x\| _{\infty}$. Here, we define $$\| x\| _1=\sum_{j=1}^{n}|x_j| \ \ \text{and} \ \ \| x\| _{\infty}=\max_{1\leq j\leq n} |x_j|.$$ I am having difficulty proving this inequality, as I don't understand how the sum of all components of a vector $x\in\mathbb{R}^n$ can be less than $n$ times the largest component in absolute value.

To me, this seem very intuitive. The sum of all components must be greater than the single largest component.

Any advice is appreciated.

Answer 1

Simple:

$\begin{align*} \| x \|_1 &= \sum_{1 \le k \le n} \lvert x_k \rvert \\ &\le \sum_{1 \le k \le n} \| x \|_\infty \\ &= n \| x_k \|_\infty \end{align*}$

Second step is from the definition as the maximum of the $x_k$.