Given a sequence $(X_{n})$ which is Cauchy in Probability sense i.e. $\forall \epsilon > 0, \exists N_{\epsilon}$ s.t.
$P\{|X_n - X_m|>\epsilon \}<\epsilon$ $\forall n,m>N_\epsilon$. In the first part to a question, I proved $\exists$ a subsequence and a random variable $X$ s.t. $X_{{n}_{k}}\rightarrow X$ a.s. using BC lemma.
But I am having trouble showing $X_n\rightarrow X$ in probability? My intuition says if for every subsequence of $X_n$ there is a further subsequence which converges to $X$, then I can make a case for $X_n\rightarrow X$ in probability. But in the previous case, I proved existence of a particular subsequence and am not sure how to make a connection to every subsequence. Any hints are appreciated Thanks.