analysis - Showing that a sequence converges (in metric space)

In $(\ell ^\infty,{\Vert .\Vert_\infty)}$, how would I show that $x_n=\left(\frac{n+1}{n},\frac{n+2}{2n},\frac{n+3}{3n}, ...\right)$ converges and how would I find the limit?

I tried using the fact that the uniform norm ${\Vert .\Vert_\infty}= \text{sup}|X_n|$ and the definition of convergence is that given $\epsilon > 0$, there exists $N \in \mathbb N$ such that $d(x_n,x)< \epsilon$ for all $n>N$, but I cant seem to show it converges. How would I show it converges and find the limit?

Answer

We denote $x^{(n)}$ the sequence of sequences. Then $x_k^{(n)}=\frac{n+k}{kn}=\frac 1k+\frac 1n$, and we can write $x^{(n)}=a^{(n)}+b^{(n)}$ where $a^{(n)}_k=\frac 1k$ for all $n$ and $b_k^{(n)}=\frac 1n$. We have $||b^{(n)}||_{\infty}=\frac 1n$ which converges to $0$ and $a^{(n)}$ doesn't depend on $n$. Denoting this sequence $a$, we can see that $\lim_{n\to\infty}||x^{(n)}-a||_{\infty}=\lim_{n\to\infty}||b^{(n)}||_{\infty}=0$.