Editing Infinite monkey theorem (section)

====Infinite strings====
This can be stated more generally and compactly in terms of [[string (computer science)|strings]], which are sequences of characters chosen from some finite [[alphabet]]:
* Given an infinite string where each character is chosen independently and [[Uniform distribution (discrete)|uniformly at random]], any given finite string almost surely occurs as a [[substring]] at some position.
* Given an infinite sequence of infinite strings, where each character of each string is chosen independently and uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

Both follow easily from the second [[Borel–Cantelli lemma]]. For the second theorem, let ''E''<sub>''k''</sub> be the [[event (probability theory)|event]] that the ''k''th string begins with the given text. Because this has some fixed nonzero probability ''p'' of occurring, the ''E''<sub>''k''</sub> are independent, and the below sum diverges,
:<math>\sum_{k=1}^\infty P(E_k) = \sum_{k=1}^\infty p = \infty,</math>
the probability that infinitely many of the ''E''<sub>''k''</sub> occur is 1. The first theorem is shown similarly; one can divide the random string into nonoverlapping blocks matching the size of the desired text and make ''E''<sub>''k''</sub> the event where the ''k''th block equals the desired string.{{efn|The first theorem is proven by a similar if more indirect route in Gut (2005).<ref>{{cite book |last=Gut |first=Allan |title=Probability: A Graduate Course |year=2005 |publisher=Springer |isbn=0-387-22833-0 |pages=97–100}}</ref>}}