transformation in normed spaces and cauchy sequence

Question

Let $T: U\rightarrow W$ a linear transformation between normed linear spaces. Show that T is continuous if and only if T maps cauchy sequences of U into cauchy sequences in W.

Can resolve seeing the cauchy sequences as bounded sequences? or $\left \| T(x) \right \|\leq k\left \| x \right \|$ for some positive k?

Answer 1

(1). If $\|T\|=\sup \{\|T(x)\|/\|x\|:x\ne 0\}<\infty$ then $T$ is Lipschitz-continuous and therefore maps Cauchy sequences to Cauchy sequences.

(2). If $\infty=\sup \{\|T(x)\|/\|x\|:x\ne 0\}$ then take a sequence $(x_n)_n$ in $U$ where $\|x_n\|=1$ and $\|T(x_n)\|>n^2$ for all $n$. Then $(y_n)_n=(x_n/n)_n$ converges to $0$, but $\|T(y_n)\|>n$, so $(T(y_n))_n$ is not a Cauchy sequence and $T$ is discontinuous.