Proof of the Triangle Inequality .
Proof of the reverse triangle inequality:
Answer & Explanation
Move |x| to the right hand side in the first inequality and |y| to the right hand side in the second inequality. We receive
From absolute value properties, we have that , and if .
Putting these two facts together, we get the reverse triangle inequality:
Consider . Therefore:
Since we are thinking about the reals, R, then a field's axioms apply. specially for.
Then apply . By so-called "first triangle inequality."
The item of Analysis that I find the most conceptually daunting at times is the notion of order , note how some sentences might be improved into shorter versions.
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