Example 3.1.5:

The Fibonacci numbers are recursively defined as x1 = 1, x2 = 1, and for all n > 2 we set xn = xn - 2 + xn - 1. The sequence of Fibonacci numbers {1, 1, 2, 3, 5, ...} does not converge.
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We will show by induction that the sequence of Fibonacci numbers is unbounded. If that is true, then the sequence can not converge, because every convergent sequence must be bounded.

As for the induction process: The first terms of the Fibonacci numbers are

{1, 1, 2, 3, 5, 8, 13, 21, ...}
We will show that the n-th term of that sequence is greater or equal to n, at least for n > 4.
Property Q(n):
xn n for all n > 4
Check Q(5) (the lowest term):
x5 = x4 + x3 = 3 + 2 = 5 5 is true.
Assume Q(n) true:
xn n for all n > 4
Check Q(n+1):
xn + 1 = xn + xn - 1 n + xn - 1 n + 1 n
Hence, by induction the Fibonacci numbers are unbounded and the sequence can not converge.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007