## 3.1. Sequences

### Example 3.1.5:

The Fibonacci numbers are recursively defined as

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.
*x*,_{1}= 1*x*, and for all_{2}= 1*n > 2*we set*x*. The sequence of Fibonacci numbers_{n}= x_{n - 2}+ x_{n - 1}*{1, 1, 2, 3, 5, ...}*does not converge.As for the induction process: The first terms of the Fibonacci numbers are

We will show that the{1, 1, 2, 3, 5, 8, 13, 21, ...}

*n*-th term of that sequence is greater or equal to

*n*, at least for

*n > 4*.

**Property Q(n)**:-
*x*for all_{n}n*n > 4* **Check Q(5)**(the lowest term):-
*x*is true._{5}= x_{4}+ x_{3}= 3 + 2 = 5 5 **Assume Q(n) true:**-
*x*for all_{n}n*n > 4* **Check Q(n+1):**-
*x*_{n + 1}= x_{n}+ x_{n - 1}n + x_{n - 1}n + 1 n