If either $X$ or $Y$ is reflexive, then every bounded linear operator $T:X \rightarrow Y$ is weakly compact

Question

I am trying to prove the previous statement, but I am stuck in some steps. My attemp so far:

-If $Y$ is reflexive, since $T(B_X)$ is bounded, it must be relatively weakly compact because

Claim: In a reflexive Banach space any bounded set is relatively weakly compact

But I cannot prove this claim!

-On the other hand, if X is reflexive, $B_X$ is weakly compact, so $T(B_X)$ is weakly compact. But how can I prove that in fact it is also relatively weakly compact?

Thanks in advance.