Here is the question:
Given $X$ and $Y$ are Banach spaces and $T_n$ : $X$ → $Y$ a sequence of bounded operators.
Show that the sequence $(|f(T_nx)|)$ is bounded for every x ∈ $X$ and every f ∈ $Y^∗$ implies the sequence $(T_nx)$ is bounded for every x ∈ $X$.
Here is the given proof:
I would like know how to show $\|g_n\|$ $\geq$ $\|x_n\|$ in the last line. This is the only point where I get stuck.