I know Cauchy sequences are only guaranteed converge in complete metric spaces.
However, I have been struggling with one issue. It seems that every Cauchy sequence converges, at least in a larger metric space (using the same distance function). Is the following conjecture true or false? (I'm almost sure it is false, but I cannot find a counter-example).
Conjecture: Let $X_{n}$ be a Cauchy sequence in an incomplete metric space $(M,d_M)$. Suppose $X_{n}$ does not converge in $(M,d_M)$. Then, there exists a metric space $(N,d_M)$, where $N \supseteq M$ , endowed with the same distance function $d_M$ such that $X_{n}$ converges in $(N,d_M)$.
I think you can say this is true.
Lets say that cauchy sequence $\{a_i\}$ in $M$ is related to cauchy sequence $\{b_i\}$ in $M$ if $d_M(a_i,b_i)\to 0$. We can prove that is an equivalence relation (I think). So if $N=$ the set of all equivalence classes of cauchy sequences then if we consider each $m \in M$ as equal to the equivalence class of all cauchy sequences that converge to $m$ then $M \subset N$ and $M$ is complete.