Balls and vase $-$ A paradox? Question
I have infinity number of balls and a large enough vase. I define an action to be "put ten balls into the vase, and take one out". Now, I start from 11:59 and do one action, and after 30 seconds I do one action again, and 15 seconds later again, 7.5 seconds, 3.75 seconds...
What is the number of balls in the vase at 12:00?
My attempt
It seems like that it should be infinity (?), but if we consider the case:
Number each balls in an order of positive integers. During the first action, I put balls no. 1-10 in, and ball no.1 out, and during the $n^{\text{th}}$ action I take ball no. $n$ out. 
In this way, suppose it is at noon, every ball must have been taken out of the vase. So (?) the number of balls in the vase is
Zero???
My first question: if I take the ball randomly, what will be the result at noon? (I think it may need some probability method, which I'm not familiar enough with.)
Second one: is it actually a paradox?
Thanks in advance anyway.

What you just discovered is that the cardinality of a set (the number of elements) is not a continuous function, that is, for a convergent sequence $S_n$ of sets you may have
$$\lim_{n\to\infty}\left|S_n\right|\ne \left|\lim_{n\to\infty} S_n\right|$$
where $\left|S_n\right|$ is the cardinality of $S_n$ (e.g. $\left|\{2,3,4,5\}\right|=4$). In your case, the left hand side diverges (giving you the infinite number of balls), while the right hand side gives $0$ (the cardinality of the empty set).
This is not a paradox, but a warning that you have to be careful with such limits.
