Prove that ||x|−|y||≤|x−y|||x|-|y||\le |x-y|

I’ve seen the full proof of the Triangle Inequality
|x+y||x|+|y|.
However, I haven’t seen the proof of the reverse triangle inequality:
||x||y|||xy|.
Would you please prove this using only the Triangle Inequality above?

Thank you very much.

Answer

|x|+|yx||x+yx|=|y|

|y|+|xy||y+xy|=|x|

Move |x| to the right hand side in the first inequality and |y| to the right hand side in the second inequality. We get

|yx||y||x|

|xy||x||y|.

From absolute value properties, we know that |yx|=|xy|, and if ta and ta then t|a|.

Combining these two facts together, we get the reverse triangle inequality:

|xy|||x||y||.

Attribution
Source : Link , Question Author : Anonymous , Answer Author : J. W. Tanner

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