Completeness of a metric space

Question

I need help solving the following exercise:

Show that a metric space $(X, d_X)$ is complete if, and only if, for every isometric embedding $f:X \to Y$ in another metric space $(Y,d_Y)$, it holds that $f(X)$ is complete in $Y$.

Answer 1

Suppose that $X$ is complete, let $f:X\rightarrow Y$ be an isometry and $(y_n)$ a Cauchy sequence of $f(X)$. Write $y_n=f(x_n)$. For every $c>0$, there exists $N$ such that $n,m>N$ implies that $d(y_n,y_m)<c$. We have $c>d(y_n,y_m)=d(f(x_n),f(x_m))=d(x_n,x_m)$ since $f$ is an isometry. We conclude that $(x_n)$ is a Cauchy sequence which converges towards an element $x$ since $X$ is complete; $lim_nd(y_n,f(x))=lim_nd(f(x_n),f(x))=lim_nd(x_n,x)=0$. Thus $(y_n)$ converges towards $f(x)$.

Suppose that for every isometry $X\rightarrow Y$, $f(X)$ is complete. The identity $Id_X:X\rightarrow X$ is an isometry, thus $(id_X(X)=X,d)$ is complete.