Let $X$ be a Banach space and $X^*$ its dual space.
a) If $\left\lbrace x_n\right\rbrace$ converges weakly to $x$ in $X$, then $\sup_n \|x_n\| < \infty$ and $\liminf_n \|x_n\| \geq \|x\|$
b) If $\left\lbrace f_n\right\rbrace$ converges weakly-* to $f$ in $X^*$, then $\sup_n \|f_n\| < \infty$ and $\liminf_n \|f_n\| \geq \|f\|$
I believe I was able to prove the first part of $a$, but I'm not sure how to prove the second part at all. I know that since $x_n \rightarrow x$ weakly, then for all $f \in X^*$, $f(x_n) \rightarrow f(x)$. According to my notes from class (and we rushed through this part very quickly):
For every $x \in X$, there exists an $f \in X^*$ such that $\|f\|=1$ and $f(x) = \|x\|$ (why is this?). Then $\|x\| = f(x) = \lim_n f(x_n) = \liminf_n f(x_n) \leq \liminf_n \|f\|\|x_n\| = \liminf \|x_n\|$. Is that right?
For part $b$, is there a slick way to use the results from part $a$ to quickly conclude the results in part $b$?