Example:
The Fibonacci numbers are recursively defined as
x0 = 1,
x1 = 1, and for all n > 1
we set
xn = xn - 2 + xn - 1.
The sequence of Fibonacci numbers {1, 1, 2, 3, 5, ...} does not
converge.
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.
- Property Q(n):
-
xn
n for all
n > 4
- Check Q(5):
-
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
Hence, by induction the Fibonacci numbers are unbounded, and the
sequence can indeed not converge.
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