Proof/Explanation to why bullet hits the apple

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KingOfQuestions
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I have a bullet with a initial velocity and an apple in free fall (-9.8 acceleration). Whenever I shoot the bullet it will hit the apple as long as the vector of the bullet is aimed at the apple. What I want to know is why does that occur? Why is it that as long as I aim the apple, the bullet hits it?
 
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No this is not really a home-work question? It's just a little question that I had. But I really need to know the answer.
 
KingOfQuestions said:
I have a bullet with a initial velocity and an apple in free fall (-9.8 acceleration). Whenever I shoot the bullet it will hit the apple as long as the vector of the bullet is aimed at the apple. What I want to know is why does that occur? Why is it that as long as I aim the apple, the bullet hits it?
The top rated comment by CGP Grey under this video has a nice graphics explaining that:

 
The bullet and the apple fall at the same rate.

Imagine performing the experiment in space, let the apple go and simultaneously fire the gun at it from a range of 300 m with a bullet velocity of 300 m/s, the bullet will hit the apple at the apples original location 1 second later.

Now modify the experiment by adding the earth. 1 second after being dropped the apple will be 4.9 m below its original position, and the bullet will be 4.9 m below it's original trajectory for exactly the same reason. Gravity accelerated them both by the same amount for the same time, and so displaced them by the same distance.