Is the sequence converging?

Question

Consider the sequence x1, x2, x3 ... defined by

Xn={3+(1/n)} (if n is odd) and {4+(1/n)} (if n is even)

As n→∞, the value of the first sequence will be 3 and the second one will be 4. Does this mean that the sequence isn't converging (since the sequence is reaching two different values)? and would I be right to use the "Proof of Cauchy sequence method" as proof of convergence?

The Cauchy Sequence Method being: for all Ɛ > 0 we can find a positive integer (N) such that for all n,m>N the distance between the terms get smaller ǀXn – Xmǀ < Ɛ

Answer 1

Two subsequences converging to different limits is enough (to conclude that the sequence does not converge), but if you can't use this or prefer to use the Cauchy criterion, that's possible too.

The Cauchy Sequence Method being: for all Ɛ > 0 we can find a positive integer (N) such that for all n,m>N the distance between the terms get smaller ǀXn – Xmǀ < Ɛ

Hint: for $n$ sufficiently large, you can get the odd terms arbitrary close to $3$ and the even terms arbitrary close to $4$. Get them close enough, to $3$ and $4$ respectively, so that you know (for sure) that the distance between those terms (from there on) will be bigger than some $\varepsilon$.