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Math Help - twist on the triangle inequality proof

  1. #1
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    twist on the triangle inequality proof

    i'm stuck on this one:

    Prove that || z1|-|z2|| |z1-z2|

    Thanks!
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  2. #2
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    The usual triangle inequality tells you that |z_1| = |z_2 + (z_1-z_2)|\leqslant|z_2| + |z_1-z_2|, and hence |z_1|-|z_2|\leqslant|z_1-z_2|. The same inequality with z_1 and z_2 interchanged gives |z_2|-|z_1|\leqslant|z_2-z_1| = |z_1-z_2|. The two inequalities together give \bigl||z_1|-|z_2|\bigr|\leqslant|z_1-z_2|.
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