Linear Operator norm for Banach Steinhaus proof

Question

As a prerequisite for the proof of the Banach-Steinhaus theorem, the following is used:

For two normed vector spaces $E,F$ and a continuous linear mapping $T: E\rightarrow F$, for all $x_0\in E,r\ge0$ it holds that $$\sup_{||x-x_0||_E\le r} ||T(x)||_F\ge ||T||r$$

I can't find the solution, proofs of the theorem mostly use that $$r||T(x)||= ||T(x_0+rx) - T(x_0)||\le ||T(x_0 + rx)|| + ||T(x_0)||$$ from where $$\sup_{||x-x_0||_E\le r} ||T(x)||_F=\sup_{||x||_E\le1}T(x_0+rx)$$ almost does it, which is sufficient for Banach Steinhaus, but that is not strong enough for the statement.

Answer 1

I suggest doing exactly what you are doing, but begin with $$ 2r \| T(x) \| = \| T(x_0 +rx) - T(x_0 - rx)\|. $$