The infinite monkey theorem is straightforward to prove, even without appealing to more advanced results. If two events are statistically independent, meaning neither affects the outcome of the other, then the probability of both happening equals the product of the probabilities of each one happening on its own. Suppose a typewriter has 50 keys, and the word to be typed is "banana". Typing at random, the chance that the first letter typed is b is 1/50, as is the chance that the second letter typed is a, and so on. These events are independent, so the chance of the first six letters matching "banana" is (1/50)^6. For the same reason, the chance that the next 6 letters match "banana" is also (1/50)^6, and so on.
Now, the chance of not typing "banana" in each block of 6 letters is 1 - (1/50)^6. Because each block is typed independently, the chance, X, of not typing "banana" in any of the first n blocks of 6 letters is X = (1 - (1/50)^6)^n. As n grows, X gets smaller. For an n of a million, X is 99.99%, but for an n of 10 billion X is 53% and for an n of 100 billion it is 0.17%. As n approaches infinity, the probability X approaches zero; that is, by making n large enough, X can be made as small as one likes. If we were to count occurrences of "banana" that crossed blocks, X would approach zero even more quickly. The same argument applies if the monkey were typing any other string of characters of any length.
The same argument shows why infinitely many monkeys will (almost surely) produce a text as quickly as it would
be produced by a perfectly accurate human typist copying it from the original. In this case X = (1 - (1/50)^6)^n where X represents the probability that none of the first n monkeys types "banana" correctly on their first try. When we consider 100 billion monkeys, the probability falls to 0.17%, and as the number of monkeys n increases to infinity the value of X (the probability of all the monkeys failing to reproduce the given text) decreases to zero. This is equivalent to stating that the probability that one or more of an infinite number of monkeys will produce a given text on the first try is 100%, or that it is almost certain they will do so.