How can we tell whether a sequence of numbers converges—that is, whether it draws ever closer to a fixed limit? If l is that limit and *un is this sequence, we can consider the difference un – l and try to prove that it approaches 0 as n* grows.
Of course, we must then explain what we mean by "approaching 0", but in many cases this can be done reasonably simply and convincingly.
For example, this method shows that the sequence with general term (n + 1)/n tends to 1 as n tends to infinity: we have (n + 1) / n – 1 = 1/n, and this last expression tends to 0 as n grows ever larger.
