Answer
Yes, it's necessarily true that player A has a higher batting average
than player B for the entire season.
Work Step by Step
Lets consider the batting average of player B for the first half of the season is $X$; so the batting average of player A for the first half of the season is at least $X+1$
And the same with the second half of the season, player B has average equal $Y$ and player A has average equal $Y+1$
So; the averages for the entire season for both players are
$(X+Y)\div2$ for player B
And $(X+1+Y+1)\div2$ which equal $(X+Y)\div2+1$ for player A
So, we find that it's necessarily true that player A has a higher batting average than player B for the entire season.
