In this paper, we prove the celebrated Banach contraction mapping theorem and a result of Mustafa and Obiedat in a G-metric space using only elementary properties of greatest lower bound. This idea of using greatest lower bound properties in metric space was ini-tiated by Joseph and Kwack in 1999. Also we introduce the notion of G-contractive fixed point and demonstrate that the unique fixed point will be a G-contractive fixed point for the underlying self-map in both the results. Our proof is highly distinct in repeatedly employing the rect-angle inequality of the G-metric rather than using traditional iterative procedure.