Is the set of all bounded sequences complete? Let $X$ be the set of all bounded sequences $x=(x_n)$ of real numbers and let $$d(x,y)=\sup{|x_n-y_n|}.$$ I need to show that $X$ is a complete metric space.
I need to show that all Cauchy sequences are convergent.
I appreciate your help.

HINT: Let $\langle x^n:n\in\Bbb N\rangle$ be a Cauchy sequence in $X$. The superscripts are just that, labels, not exponents: $x^n=\langle x^n_k:k\in\Bbb N\rangle\in X$. Fix $k\in\Bbb N$, and consider the sequence 
$$\langle x^n_k:n\in\Bbb N\rangle=\langle x^0_k,x^1_k,x^2_k,\dots\rangle\tag{1}$$
of $k$-th coordinates of the sequences $x^n$. Show that for any $m,n\in\Bbb N$, $|x^m_k-x^n_k|\le d(x^m,x^n)$ and use this to conclude that the sequence $(1)$ is a Cauchy sequence in $\Bbb R$. $\Bbb R$ is complete, so $(1)$ converges to some $y_k\in\Bbb R$. Let $y=\langle y_k:k\in\Bbb N\rangle$; show that $y\in X$ and that $\langle x^n:n\in\Bbb N\rangle$ converges to $y$ in $X$.
