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We show that `s` := [(-1)^{i}]_{i} is divergent. Assume
that `s` is convergent, i.e., `s` converges to some limit `a`
in **R**. We show a contradiction.

Let epsilon := 1/2. There exists (by definition of convergence)
some `n` in **N** such
that
**forall** `i` >= `n`: |(-1)^{i} - `a`| <
1/2.
We thus have

We then have (using the absolute value laws)

|1 - a| < 1/2 /\ |-1 -a| < 1/2

1 = 1/2+1/2 > |1 - a| + |-1 -a|= |1 - a| + |1 +a| >= |(1 -a) + (1 +a)| = 2

which represents a contradiction.

Author: Wolfgang Schreiner

Last Modification: December 14, 1999