Why is Completeness and Compactness not equivalent in Normed Spaces? Given a complete normed space $X=(X,\|\cdot\|)$.  Every Cauchy sequence converges in it. I am not able to understand why we can't show that every bounded sequence in $X$ will have a convergent subsequence.
Please give an example to clarify why completeness does not imply compactness and do explain where does the problem lie.

Your question seems to be about the local compactness of Banach spaces.
Look at a sequence of sequences with general term $A_n=(\delta_{i,n})_{i>0}$ which is in $l^p$. What can you say about the norm of its term and the norm of the "potential" limit? By the way there is compactness in a weak sense, but you might need Hilbert spaces for their scalar product. See Banach-Alaoglu's theorem.
It is also worth mentioning that a general result exists: A normed vector space (real or complex)'s closed unit ball is compact (for the norm topology) if and only if it is of finite dimension (Riesz lemma).
