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The difference of norms is bounded by the norm of the difference
Reverse triangle inequality: In a normed vector space , for all ,
Equivalently,
This inequality is frequently used to show continuity of the norm map and to transfer convergence of vectors to convergence of their norms.
Let be a function between metric spaces, and let . We say is continuous on if is continuous at every point (equivalently, the restriction is continuous).
Spelled out: for every and every , there exists such that for all ,
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
The triangle inequality states that for any three points in a metric space :
This is one of the defining axioms of a metric.