Theorem (Reverse triangle inequality): For any \(\x, \y \in \real^d\),
\[\norm{\x} - \norm{\y} \leq \norm{\x-\y}\]Corollary: For any \(\x, \y \in \real^d\),
\[\begin{align*} \norm{\x} - \norm{\y} &\leq \norm{\x+\y} \\ \norm{\y} - \norm{\x} &\leq \norm{\x+\y} \\ \norm{\y} - \norm{\x} &\leq \norm{\x-\y}. \end{align*}\]Proof: Theorem:
\[\begin{alignat*}{2} \norm{\x} &= \norm{\x - \y + \y} &\hspace{8em}& \\ &\leq \norm{\x - \y} + \norm{\y} && \href{norm.html}{\text{Triangle inequality}} \end{alignat*}\]Same idea works for corollary:
\[\begin{alignat*}{2} \norm{\x} &= \norm{\x + \y - \y} &\hspace{8em}& \\ &\leq \norm{\x + \y} + \norm{-\y} && \href{norm.html}{\text{Triangle inequality}} \\ &\leq \norm{\x + \y} + \norm{\y} && \href{norm.html}{\text{Homogeneity}} \end{alignat*}\]And so on.