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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.

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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.2012-04-05
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    A [similar](http://math.stackexchange.com/questions/127372/reverse-triangle-inequality-proof) question.2012-04-05

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