$\{T(x_n)\}$ is Cauchy in $Y$ whenever $\{x_n\}$ is Cauchy in $X$, we need to show $T$ is continuous.

Question

$X,Y$ be complex norm linear space which are not necessarily complete, let $T:X\rightarrow Y$ be a linear map such that $\{T(x_n)\}$ is Cauchy in $Y$ whenever $\{x_n\}$ is Cauchy in $X$, we need to show $T$ is continuous.

My Attempt: construct two new sequence $y_n,z_n$ such that $$y_1=T(x_1)$$ $$z_1=T(x_2)$$ $$y_2=T(x_3)$$ $$z_2=T(x_4)$$ $$\dots$$ $$y_n=T(x_{2n-1})$$ $$z_n=T(x_{2n})$$ let $x_n\rightarrow c\in X$ we need to show $T(x_n)\rightarrow T(c)\in Y$, am I going in right path?if not please help!

Answer 1

For given $x\in X$, let $\{ x_n\}$ be a sequence convergent to $x$. Define $y_n$ by setting $y_n=x_{n/2}$ when $n$ is even and $y_n=x$ otherwise. Again we have $y_n\rightarrow x$ and also $\{y_n\}$ is Cauchy. Therefore $T(y_n)$ is Cauchy which easily implies that $T(x_n)\rightarrow T(x)$.