## 3.1. Sequences

### Example 3.1.3(b):

Note that
*
= {-1, 1, -1, 1, -1, 1, ...}*.
While this sequence does exhibit a definite pattern, it does not
get close to any one number, i.e. it does not seem to have a
limit. Of course we must prove this statement, so we will use a proof
by contradiction.

Suppose that the sequence did converge to a limit *L*. Then, for
*= 1/2* there exists
a positive integer *N* such that

for all| (-1)^{n}- L | < 1/2

*n > N*. But then, for some

*n > N*, we have the inequality:

for2 = | (-1)^{n + 1}- (-1)^{n}| = | ((-1)^{n + 1}- L) + (L - (-1)^{n}) |

| (-1)^{n + 1}- L | + | (-1)^{n}- L | < 1/2 + 1/2 = 1

*n > N*, which is a contradiction since it says that

*2 < 1*, which is not true.