Infinity is a very strange concept to most people.  Most people do not ever encounter a problem in which they have to work with infinity and therefore do not have much real-life experience to relate to when encountering problems such as this.

You might be surprised to find that the first container is indeed empty.
This is why:
When you continue the process forever, is there a single number that you can say with 100% certainty will be left in the box?  The answer is no.  This may seem strange but it can be proven by finding the probability that a specific number is in the box.  This is a very complicated mathematical process to prove but here I will summarize.

 Since we are working with an infinite number of numbers, we would need to use the formula for infinite sums.  By finding the probably of any one of the numbers, since they are equally likely because each only occurs once, and using the formula of infinite sums, the sum converges to zero.

The second container is full.
This is why:
If you though about the process, couldn't you state a number that would be in the container?  As a matter of fact, there are many numbers because only the multiples of 10 will be missing. So the number 1, 2, 3, and so on, are all numbers that are in the containers. The mathematical process is the same as before except that you will find that the infinite sum of the probabilities will converge to one. So there is a 100% chance that you will find an infinite number of numbers in the container.

Strategies Home