I want to prove that a sequence is not Cauchy by proving something like $$ \exists \epsilon >0 \space s.t \space \space \forall n_0 \in N \space \space \exists n, n+p > n_0 \space=> |x_{n+p} - x_n| > \epsilon $$
For that specific question I think I can prove that by taking sequences $$x_n, x_{n+3}$$ as an example plugging that into sequence formula, however I am not sure if it is even allowed to take a specific number for the p in a proof. Can you please clarify that part?
The sequence I am trying to prove is not Cauchy is: $$x_n = {3n+1 \over n+4}sin{2\pi n \over 3}$$
Watch out:
Negating an implies symbol does NOT give another implies symbol.
That arrow should be an and symbol, or something with equivalent meaning.
You are allowed to take specific numbers for p: 3 is fine ,as if $n>n_0$, then certainly $n+3>n_0$, so that's OK!.