Why if $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that $||Tx_n||\ge n||x_n||$ If $X$ $Y$ are Banach space, $T$ is a linear operator between them.
I don't understand the following statement:
If $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that $$||Tx_n||\ge n||x_n||$$.
Why is this true? Could someone kindly explain it? Thanks!

By definition, to be continuous means there is a fixed $C>0$ s.t.
$$\Vert Tx\Vert\leq C\Vert x\Vert,\forall x\in X$$
So if $T$ is not continuous, then $\forall C>0$, $\exists x_C\in X$ (here $x_C$ depends on the choice of $C$) s.t.
$$\Vert Tx_C\Vert > C\Vert x_C\Vert$$
In particular, you can choose $C=n$ for any $n\in\mathbb{N}$, and in which case $x_n=x_C$. 
