Prove that $||a|-|b||\leq|a-b|$.

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Lesson Parent: 
Rules of Absolute Values
Problem: 

Prove that
\begin{equation}
\left| |a|-|b| \right|\leq\left| a-b\right|.
\end{equation}

Answer: 

It is true that \begin{equation} \left| |a|-|b| \right|\leq\left| a-b\right|. \end{equation}

SAMPLE SOLUTION

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Solution: 

We know that <Sign in to see all the formulas>. Let's do some algebra:

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We used the fact that <Sign in to see all the formulas> to get the last equation. Using this fact again we see that

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and therefore

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