Convergence of a sequence in metric space

Question

Can someone please help me with this problem? Thanks!

Check if the sequence $x_n= (1+1/n)^n$ is convergent in $ (X,d)$ where $d(x,y)=$ $\frac {2|x-y|}{3+2|x-y|}$, and if it is convergent, then find its limit.

Answer 1

Since that distance is topologically equivalent to the usual distance and since, with respect to the usual distance, that sequence converges to $e$, then the sequence converge to $e$ in $(\mathbb R,d)$ too.

Answer 2

let $g:[0,\infty) \to [0,\infty)$ be $g(t) = {2 t \over 3+2t}$. Note that $g(t) \le {2 \over 3} t$ and for $t \le 1$, we have $g(t) \ge {2 \over 5} t$.

Note that $d(x,y) = g(|x-y|)$. In particular, $x_n$ is convergent with respect to $g$ iff it is convergent with respect to $|\cdot|$.