Slims

You have an infinite number of marbles numbered sequentially 1,2,3, etc. and a bag to hold them all.

Problem 1:
If you put the marbles in the bag, ten at a rate of ten marbles per second, sequentially (marbles 1-10 first, 11-20 second, etc.), and you remove the marbles at a rate of 1 per second, sequentially (I.e. 1,2,3, etc.), how many marbles are in the bag as time goes to infinity?

Problem 2: Same as 1, except you remove the marbles by pulling out every tenth marble (i.e. you remove number 10, 20, 30, etc.) , as time goes to infinity, how many marbles are in the bag?

Hint: The answers are not the same.

It was a problem like this one that drove (I think I have the right mathematician) the russian mathematician Cantor crazy.

have fun.

Please feel free to post your answers/proofs.