Proof: A Useful Absolute Value Inequality | Real Analysis

For real numbers a and b, |a| is less than or equal to b if and only if -b is less than or equal to a is less than or equal to b. We'll prove this useful theorem about the absolute value in today's real analysis lesson!

This theorem gives us a way to go from an absolute value inequality (|a| is less than or equal to b) to a nice inequality with no absolute value! We can of course also use the other direction of the theorem in order to PROVE an absolute value relationship. If we want to prove that |a| is less than or equal to b, we just need to show that -b is less than or equal to a is less than or equal to b. That is the power of this result, it helps us work with absolute value! The basic idea is that if |a| is less than or equal to b, it must be the case that -b is more negative than a and b is more positive than a. Thus, whether a is positive or negative, its magnitude is less than b. Notice if b is greater than or equal to |a|, then b must be non-negative.

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