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We augment the dimension of the Euclidean space by one and the Picarditeration of a contraction by a simple iteration on the real line such that theresulting iteration becomes monotone increasing and bounded with respect to theorder defined by the Lorentz cone of the augmented space. This provides adifferent way of showing the convergence of the Picard iteration of acontraction, exhibiting the strong relationship between the Banach fixed pointprinciple and the ordering structure of the Euclidean space ordered by theLorentz cone.

The Banach fixed point principle viewed as a monotone convergence with respect to the Lorentz cone