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[Note that in my answer I am actually assuming that there are only a finite number of monkeys. I don't see what is gained by having both the number of monkeys and the time frame be infinite: mathematically speaking $\aleph_0 \times \aleph_0 = \aleph_0$, and it is somewhat confusing to contemplate infinitely many monkeys typing simultaneously: too much is happening at once. In fact, there might as well be only one monkey, or at any rate only one typewriter.]

Let me take the unusual (for me) step of considering the practical aspects of this question as well.

As Ross Millikan has explained, there is a simple mathematical model of monkey keyboard pounding under which it is easy to see that the claim is true: the probability that at least one of the monkeys will type out Hamlet approaches $1$ as the time $n$ approaches infinity.

However there is an assumption here: namely, that the pounding on the typewriter is random or sufficiently close to random. One way to formalize this is to say that after typing any $n$ characters, the probability of hitting any given key as the $n+1$st character is at least $P$, where $P$ is positive and independent of $n$.

The problem is that for actual typewriter banging, this is a very unlikely assumption. The issue is similar here to what happens if you ask someone to produce a random sequence of digits, say from $0$ to $9$, or even a random sequence of $H$'s and $T$'s (for "heads" and "tails"). Just closing your eyes and banging away will produce something very far from being random.

If the question is meant to apply to actual monkeys with their nonrandom motor behavior, then it is something else entirely. I would be tempted to say that the probability of producing Hamlet does not approach $1$ as time approaches infinity, but I'm not sure off the top of my head how to justify this.