prove if a l B and B does not equal zero then l a l < (or equal to) l b l Soluti

Question

prove if a l B and B does not equal zero then l a l < (or equal to) l b l

Explanation / Answer

i) |a| + |b| = a + b and |a + b| = a + b since a + b > 0 Thus equality holds. ii) Since a+b < 0, then |a + b| = -(a + b) = -a - b and |a| + |b| = - a - b so again equality holds. iii) a + b > 0 hence |a + b| = a + b But |a| + |b| = a - b Therefore |a| + |b| - |a + b| = (a - b) - (a + b) = -2b > 0 since b < 0 Therefore |a + b| < |a| + |b| iv) a + b < 0 hence |a + b| = -(a + b) Now |a| + |b| = a - b therefore |a| + |b| - |a + b| = (a - b) + (a + b) = 2a > 0 hence, as before, |a + b| < |a| + |b|

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