Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$ and let $l^\infty_\mathbb{F}$ be the vector space of bounded sequences with the usual addition and scalar multiplication, equipped with the norm $\|\xi\|:=\sup_{j\in\mathbb{N}}|\xi_j|$. Then prove that $l^\infty_\mathbb{F}$ is complete.
Here is what I have done:
Let $\{\xi^{(i)}\}_{i\in\mathbb{N}}\subset l^\infty_\mathbb{F}$ be a Cauchy sequence. Then for $j\in\mathbb{N}$ we have that $\{\xi^{(i)}_j\}_{i\in\mathbb{N}}$ is Cauchy, thus we define $\xi^*$ by $$\xi^*_j=\lim_{i\to\infty} \xi^{(i)}_j.$$ First we have to show that $\xi^*\in l^\infty_\mathbb{F}$. By hypothesis there exists $N\in\mathbb{N}$ such that $$\|\xi^{(m)}-\xi^{(n)}\|<1 \qquad \text{whenever}\qquad m,n\ge N.$$ Thus if $n\ge N$ we have $$\|\xi^{(n)}\|\le\|\xi^{(N)}\|+\|\xi^{(n)}-\xi^{(N)}\|<\|\xi^{(N)}\|+1$$ and thus for $B:=1+\max_{1\le i\le N}\|\xi^{(i)}\|$ we see that $$\|\xi^{(n)}\|<B \qquad \text{for all}\qquad n\in\mathbb{N}.$$ Suppose that there exists $j\in\mathbb{N}$ with $|\xi^*_j|>B+1$. Then by construction there exists $K\in\mathbb{N}$ such that $$|\xi^*_j-\xi^{(k)}_j|<1 \qquad \text{whenever}\qquad k\ge K$$ and thus $$|\xi^{(K)}_j|\ge |\xi^*_j|-|\xi^*_j-\xi^{(k)}_j|>B$$ which is a contradiction. Thus $\xi^*$ is bounded i.e. $\xi^*\in l^\infty_\mathbb{F}$. Now I'm not able to show that $$\xi^{(i)}\overset{i\uparrow\infty}{\to}\xi^*$$ because I don't see how to control $|\xi^*_j-\xi^{(i)}_j|$ for infinitely many $j$ at once. How to do it?
Please give only hints.
Thanks to the comment of xyzzyz I found the following proof. Let $\varepsilon>0$, then there exists $N\in\mathbb{N}$ such that $$ \|\xi^{(m)}-\xi^{(n)}\|<\frac{\varepsilon}{4} \qquad\text{whenever}\qquad m,n\ge N. $$ Furthermore there exists $j\in\mathbb{N}$ such that $$ \|\xi^{*}-\xi^{(N)}\|-\frac{\varepsilon}{4}<|\xi^{*}_j-\xi^{(N)}_j\|\le\|\xi^{*}-\xi^{(N)}\|. $$ Finally there exists $M\in\mathbb{N}$ such that $$ |\xi^{*}_j-\xi^{(m)}|<\frac{\varepsilon}{4} \qquad\text{whenever}\qquad m\ge M. $$ Let $m=\max\{M,N\}$. Thus, if $n\ge N$ we have $$ \|\xi^{*}-\xi^{(n)}\|\le\|\xi^{*}-\xi^{(N)}\|+\|\xi^{N}-\xi^{(n)}\|<|\xi^{*}_j-\xi^{(N)}_j|+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}\le|\xi^{*}_j-\xi^{(m)}_j|+|\xi^{(m)}_j-\xi^{(N)}_j|+\frac{\varepsilon}{2}<\\\frac{\varepsilon}{4}+\|\xi^{(m)}-\xi^{(N)}\|+\frac{\varepsilon}{2}<\frac{\varepsilon}{4}+\frac{\varepsilon}{4}+\frac{\varepsilon}{2}=\varepsilon $$ And thus $$ \lim_{n\to\infty}\xi^{(n)}=\xi^*. $$