Unbounded linear operator between normed spaces

Question

I am in the middle of a proof and this is one step I don’t understand

Let $T:E\rightarrow F$ be a linear operator between normed spaces $E$ and $F$

If $T$ is unbounded then there exists a sequence $(x_n)$ in $E$ such that $0<||x_n||\leqslant 1$ and $||Tx_n||\geqslant n$ for all $n \in \mathbb{N}$

Is this trivial. I just don’t see how we can just pick such a sequence.

Answer 1

Since $||T||=\sup_{||x||= 1} ||T(x)||$, for every natural number $n$ there exist an $x_n$ so that $||x_n||= 1$ and $||T(x_n)||\geq n$.