How do I show that $\ell^{\infty}$ is a normed linear space? How do I show that $\ell^{ \infty}$ is a normed linear space,
where $\ell^{ \infty}$ is define as $$\|\{a_n\}_{n=1}^{\infty}\|_{\ell^\infty}=\sup_{1 \leq k \leq \infty} |a_k|?$$
There are three properties that I need to check in order for this to be a normed linear space. Nonnegativity, positive homogeneity, and the triangle inequality.
I am having trouble working with the supremum. Any idea will be greatly appreciated thanks
For the triangle inequality, suppose that $\{a_n\},\{b_n\}\in \ell^{\infty}$. For each index $n$ we have
$$|a_n+b_n|\leq |a_n|+|b_n|\leq \sup_{m}|a_m|+\sup_m|b_m|=\|a\|_{\infty}+\|b\|_{\infty}.$$
Then taking the supremum over all $n$ shows that
$$ \|a+b\|_{\infty}\leq \|a\|_{\infty}+\|b\|_{\infty}. $$
Let $(a_n),(b_n) \in \ell^\infty$ Then we have for each $k$:
$$|a_k+b_k| \le |a_k|+|b_k| \le ||(a_n)||_\infty+||(b_n)||_\infty.$$
This shows that $(a_n+b_n) \in \ell^\infty$ and that
$$||(a_n+b_n)||_\infty \le ||(a_n)||_\infty+||(b_n)||_\infty.$$