Inequality regarding weak-* convergence Let $X$ be a normed linear space, $\psi \in X^{*}$ and $\displaystyle \\{\psi_n\\}_{n \in \Bbb N}$ a sequence in $X^{*}$. Show that if $\displaystyle \\{\psi_n\\}_{n \in \Bbb N}...--prophetes.ai

Inequality regarding weak-* convergence Let $X$ be a normed linear space, $\psi \in X^{}$ and $\displaystyle \\{\psi_n\\}_{n \in \Bbb N}$ a sequence in $X^{}$. Show that if $\displaystyle \\{\psi_n\\}_{n \in \Bbb N}$ converges weak-${}$ to $\psi$ then: $$\|\psi\| \le \lim \sup \|\psi_n\|$$ Any suggestions or help will be greatly appreciated. I am not well acquainted with weak-$$ convergence. Thank you in advance!

Weak-$*$-convergence gives you

$$\|\psi(x)\| \le \lim \sup \|\psi_n(x)\|$$

for every $x\in X$. Can you proceed from here?