Let E be a Banach space.
Let $(x^∗_n )$ be a sequence in $E^∗$ verifying $(<x^∗_n , x>)$ converges for any $x ∈ E$.
Prove that $\exists x^∗ ∈ E^∗: (x^∗_n )$ converges vers $x^*$ for the weak-∗ topology.
The solution I have states that it is a corollary of the Banach Steinhaus theorem but I don't see how it is related and I am not aware of such a corollary.
Many thanks for your help.