$l_p$ norm and Banach

Question

I know that $l_p$ norm is $$\|x\| = ( x_1^p + \cdots + x_n^p )^{1/p}$$ and Banach space is complete normed.

But, I can not prove that the vector space $\mathbb{R}^n$ , $n \ge 2$ , with the $l_p$ norm is a Banach space.

Answer 1

$\bf Hint$: Consider a Cauchy-sequence $\{x_k\}_{k=1}^{\infty}$ in $\mathbb{R}^n$ with respect to the $\ell^p$-norm then for $j=1,...,n$ You have for every $\varepsilon>0$ there exists $N(\varepsilon)\in\mathbb{N}$ so that $$|(x_k)_j-(x_l)_j|\leq ||x_k-x_l||_p<\varepsilon$$ for all $k,l\geq N(\varepsilon)$ and thus $\{(x_k)_j\}_{k=1}^{\infty}$ is a Cauchy sequence for each $j=1,...,n$ in $\mathbb{R}$ with the usual metric and $\mathbb{R}$ is complete. Define $x\in\mathbb{R}^n$ by $$(x)_j=\lim_{k\rightarrow\infty}(x_k)_j$$. Can You take it from here?