no dude, if you reason like that you're basically assuming all the mass of the earth is located in one point, in the center of the earth.
this way of thinking is quite correct for calculating gravitational pull on objects far away, such as satellites or meteors, etc.... where you can see the earth as a point-mass
however if you're calculating the gravitational pull for this problem you must see the earth as a spherical mass with a certain radius instead of a point-mass....
So according to you way of thinking you say, the closer you approach the center of the earth (which you see as a center of all mass) your gravitational pull will increase.... this is correct, for your way of thinkingbut unfortunately the earths mass isn't all concentrated in one point, its a sphere, so in human words, once you go below the surface, the mass below you will keep pulling you towards the center, however the mass above you will pull you back up... still following ?
so when you reach the center of the earth you can say that in any direction, above, below, left, right, etc...you have an equal ammount of "earth" thats gonna result in equal ammounts of gravitational pull in that direction.
since this gravitational pull is equally large in all directions, your resulting gravitational pull will be zero.(gravity force pulling you down is cancelled by the force pulling you up, gravity force pulling you left is canceled out by the force pulling you right, etc....)
i hope this was clear enough, english isn't my main language and explaining stuff like this isn't easy to begin with
