3.1. Sequences

Example 3.1.3(b):

The sequence does not converge.

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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

| (-1) n- L | < 1/2
for all n > N. But then, for some n > N, we have the inequality:
2 = | (-1) n + 1 - (-1) n | = | ((-1) n + 1 - L) + (L - (-1) n ) |
      | (-1) n + 1 - L | + | (-1) n - L | < 1/2 + 1/2 = 1
for n > N, which is a contradiction since it says that 2 < 1, which is not true.
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