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If $\vec{x}, \vec{y} \in \mathbb{R}^n$. Is it always true that $ \|\vec{x} + \vec{y}\| \geq \|\vec{x}\| - \|\vec{y}\| $ ?

Any advice or proofs would be greatly appreciated.

Dave K.
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  • Yes, this is always true for $\mathbb{R}^n$ where the triangle inequality holds. Just interpret it as "the difference of the lengths of any two sides of a triangle must be less than the length of the third". You prove it using something like $\| x\| \leq \|x+y\|+\|y\|$, the equality holds when two vectors lie in the same direction. – Shuhao Cao Apr 05 '12 at 16:13
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    A [similar](http://math.stackexchange.com/questions/127372/reverse-triangle-inequality-proof) question. – David Mitra Apr 05 '12 at 16:13

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Hint: Apply the triangular inequality to $x=(x+y)+(-y)$.

Did
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  • Thanks! I don't know why I avoided this before. I think I was overlooking the obvious fact that $\|-\vec{y}\| = \|\vec{y}\|$. – Dave K. Apr 05 '12 at 16:20