<s>[INST] $|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$ where $a,b,c\in\mathbb{R}$ Show that $|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$ where $a,b,c\in\mathbb{R}$
I'd like to get an hint on how to get started. What I thought to do so far is dividing to cases to get rid of the absolute value. $(++, +-, -+, --)$
but it looks messy. I'm wondering if there is any nicer way to solve it.
Would love to hear some ideas.
Thanks in advance!
 [/INST] For all $a,b,c \in \mathbb{R}$,
\begin{align}
0 &\leq (b-c)^2\\
2bc &\leq b^2 + c^2\\
2a^2bc &\leq a^2b^2 + a^2c^2\\
a^4 + 2a^2bc + b^2c^2 &\leq a^4 + a^2b^2 + a^2c^2 + b^2c^2\\
a^2 + bc&\leq \sqrt{a^4 + a^2b^2 + a^2c^2 + b^2c^2} \label{1}\tag{1}\\
2a^2 + 2bc&\leq 2\sqrt{a^4 + a^2b^2 + a^2c^2 + b^2c^2}\\
2a^2 -2\sqrt{a^4 + a^2b^2 + a^2c^2 + b^2c^2}&\leq -2bc\\
2a^2 +b^2 + c^2 -2\sqrt{a^4 + a^2b^2 + a^2c^2 + b^2c^2}&\leq b^2 -2bc + c^2\\
(a^2 +b^2) -2\sqrt{(a^2+b^2)(a^2+c^2)} + (a^2 + c^2)&\leq b^2 -2bc + c^2\\
\end{align}
and then take the square root of both sides.
Note that it was not necessary to use absolute values the first time we took a square root $(\ref{1})$ because the right hand side is understood to be positive.

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