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Having carefully considered research areas in Functional Analysis, we chose to be particular about Banach Spaces. Banach Spaces are a very important set in real life applications. In fact, many real life problems ranging from engineering to economics and other fields have the basic properties of Banach Spaces. This work reviews the Banach fixed point theory and the convergence of sequences. Inclusively, a priori and posteriori error estimates, and makes explicit the proof of the convergent sequences on Banach fixed point theorem. The proof is divided into four (4) lemmas. This work goes through these lemmas thoroughly to establish them. Keywords: functional analysis, banach space, banach fixed point theory, convergent sequence, priori and posteriori error estimates.