If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence?
Attempt: $f: S \rightarrow T$ is a continuous function at $x_m \in S$
$\implies \forall \epsilon >0, \exists \delta >0$ such that $d_T(f(x_n),f(x_m) ) < \epsilon$ whenever $d_S(x_n,x_m)< \delta~~~~............(1)$
$X$ is a cauchy sequence $\implies \forall \delta >0, \exists k \in \mathbb N$ such that $d_S( x_n , x_m ) < \delta$ whenever $ m,n \geq k~~~~............(2)$
Substituting $(2) $ in $(1)$, we get :
$\forall \epsilon >0, \exists k \in \mathbb N $ such that $d_T(f(x_n),f(x_m) ) < \epsilon$ whenever $ m,n \geq k~~~~............(2)$
$\implies f(X)$ must also be a cauchy sequence.
Please tell me where I could have gone wrong in the attempt shown above?
Thank you for your help..