A linear map, continuous at zero, is bounded
Solution 1:
Suppose $\|x \| = a$; write $x = \frac{a}{\delta} y$, where $\|y\| = \delta$. You know that $T(x) = \frac{a}{\delta} T(y)$, and $\| T(y) \| < 1$. So $\|T(x)\|< |\frac{a}{\delta}|$.
The case where $\| x \| < a$ is similar.