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?