The norm can change by at most the norm of the change: $\abs{\norm{x}-\norm{y}}\leq\norm{x-y}$.Suggest a slogan
Let \((V,\norm{\cdot})\) be a normed vector space. For all \(x,y\in V\), \[ \abs{\norm{x} - \norm{y}} \leq \norm{x - y} . \]
By the triangle inequality, \(\norm{x}\leq\norm{x-y}+\norm{y}\) and \(\norm{y}\leq\norm{y-x}+\norm{x}\), and \(\norm{y-x} = \norm{x-y}\). Hence \(\norm{x}-\norm{y}\leq\norm{x-y}\) and \(\norm{y}-\norm{x}\leq\norm{x-y}\).
What is the reverse triangle inequality?
State the reverse triangle inequality.