Question

Prove {0,2,0,2,0,2,...} does not converge

Answer 1

Recall that a sequence
`x_n` is convergent if and only if
`x_n` is also a Cauchy sequence, which means to say that for any
`\varepsilon>0`, we can find a sufficiently large
`N` for which


`|x_m-x_n|<\varepsilon`

whenever both
`m` and
`n` exceed
`N`.

But this never happens if we choose
`m=n+1` and
`0<\varepsilon<2`; under these conditions, we have


`|x_m-x_n|=|x_(n+1)-x_n|=2\\ot<2`

Therefore
`x_n` is not a Cauchy sequence and hence does not converge.