A linear operator is continuous if and only if it maps cauchy sequences to cauchy sequences

Question

Let $A$ and $B$ be seminormed spaces, then I want to show that a linear operator $T: A \rightarrow B$ is continuous if and only if it maps cauchy sequences to cauchy sequences.

The direction "$T$ continuous $\Rightarrow$ $T$ maps cauchy sequences to cauchy sequences" is easy, because any linear continuous operator is a bounded operator.

But I'm stuck on the other direction, I'm aware that I can equivalently show that $T$ is continuous in $0$ or that $T$ is a bounded operator.

Any advice?

Answer 1

Suppose $T$ is not bounded. So, there is a sequence of elements $(x_n)$ of A with norm $1$ such that $\|T(x_n)\| > n^2$. But then the sequence $y_n = \frac{1}{n} x_n$ is a Cauchy sequence, because $\|y_n - y_{n+k}\| < \left |\frac{1}{n} + \frac{1}{n+k} \right| < \frac{2}{n}$ and $\|T(y_n)\| = \frac{1}{n} \|T(x_n)\| > n$ that is not Cauchy (because it's not bounded).