What I have been given is that all Cauchy Sequences are bounded (proven by the definition of Cauchy sequences), definition of Cauchy sequences, definition of convergent sequences.
Note that all Cauchy Sequences are in real set. Also, I don't yet know about spaces and all, so please don't use those terminologies and all. Also, please don't use subsequences and all, if you want, please specify their meanings and definitions.
Can I prove that Cauchy Sequences converge?
My try (Not at all successful):
Given that, for every $\epsilon>0$, there exists a natural number $N$, such that $$|a_m-a_n|<\epsilon \space\space\forall\space \space n,m\geq N$$
So $|a_n-L|\leq|a_n-a_m|+|a_n-L|\leq\epsilon+|a_m-L|$ where L is any real number.
I think that it can be done after that, but I just cannot get anything ahead.