Let us construct inductively a sequence of rationals that approximate . (This is not necessarily the best proof of the irrationality of but it gives an easy illustration of the technique.) We begin with , and observe that . If were , then we would have , so the " " at the end of this is our error term in the first approximation. Now suppose we have defined and in such a way that . Then set and . (The justification for this choice is that if were then would be too, as can easily be checked.) Then

Thus, we have constructed a sequence of rationals , with denominators tending to infinity, such that for every . But from this we deduce that , and therefore that . Since (as may easily be checked), this implies that tends to (at roughly the same rate as ), and therefore that is irrational.