A Minimizing Sequence Proof of the Banach Fixed Point Theorem

We present a novel proof of the Banach contraction mapping theorem based on minimizing sequences. By analyzing the set $\mathcal{A} = \{d(x, T(x)) : x \in X\}$ of point-to-image distances, we construct a sequence that converges to the unique fixed point. A key technical contribution is Lemma~\ref{lemma1}, which establishes an optimal inequality between contraction coefficients and the b-metric constant. We demonstrate applications to b-metric spaces and discuss extensions to incomplete metric spaces. Examples show the effectiveness of this approach, which provides a geometric alternative to classical methods.