Question

How do you do this?

Answer 1

See below:

Explanation:

The sequence is

`1,0,-1,0,1,0,-1,0,...`

Although it is rather clear from inspection that this sequence does not converge, a more formal proof can be given by several different approaches.

A more advanced approach would be to notice that the series has sub-sequences that converge to different limits (namely `1`, `-1`, and `0`) - and this shows that the sequence proper has no limit.

A direct approach from the definition uses a proof by contradiction. We assume that a limit `L` exists. This implies that

`\forall epsilon>0,\ exists N in NN\ :\ n>N implies |a_n-L| < epsilon`

Since `1 in {a_n| n> N}`, we have `|1-L| < epsilon`
Similarly `-1 in {a_n| n> N}implies|-1-L| < epsilon`

Thus

`2 = |1-(-1)| = |(1-L)-(-1-L)|`
`qquad <= |1-L|+|-1-L|<2 epsilon`

Thus, we end up with `forall epsilon >0, epsilon>1` - which is obviously a contradiction. Hence the limit does not exist.