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 initiated 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 rectangle inequality of the $G$-metric rather than using traditional iterative procedure.