Answer
Yes.
Work Step by Step
Let $m$ and $n$ be the batting averages for players A and B for the first half respectively. It is given that $m>n$.
Now let $f$ and $g$ be the batting averages for players A and B for the second half respectively. It is also given that $f>g$.
If $a>b$ and $c>d$, then it is also true that $a+c>b+d$. We will apply this in this situation. $m+f>n+g$. We will find the averages and show that player A must have a higher batting average overall.
$m>n$, $f>g$
$m+f>n+g$
$\frac{m+f}{2}>\frac{n+g}{2}$
This is true for all cases.
