Inequality for $l^p$-norm

Question

I am having troubles to show that $\|x\|_{\infty}\leq\|x\|_{p}$ is true for all $x\in l^{p}$ with $1\leq p <\infty$.

I have only so far \begin{equation} \|x\|_{\infty}=\sup_{n \in\mathbb{N}}|x_n|\leq\sum_{n=1}^{\infty}|x_n|, \end{equation} which is true because the supremum is included on the sum and every term is non-negative. My problem is to complete the full form of the $p$-norm $\|x\|_{p}=(\sum_{n=1}^{\infty}|x_n|^p)^{1/p}$ on the right hand side of the inequality. I have tried to use the Hölder's inequality like here https://en.wikipedia.org/wiki/Lp_space#Embeddings but in my case is $q=\infty$. I really apreciate your answers.

Answer 1

For $\epsilon > 0$ let $n\in\mathbb N$ such that : $$\left\|x\right\|_\infty - \epsilon \le \left|x_n\right|$$

So $$\left(\left\|x\right\|_\infty - \epsilon\right)^p \le \left|x_n\right|^p \le \sum_{k=0}^\infty \left|x_k\right|^p = \left\|x\right\|_p^p.$$

Taking the $1/p$ exponent et $\epsilon \to 0$ you have your result.