Prove that for side-lengths of a triangle then prove that the following inequality holds ,

*Note : The inequality as proposed originally holds for all positive reals….but *

I amn’t good enough to solve it without the extra condition I have imposed.

**Proof**

Lemma 1

For all positive reals ,

Lemma 2

For all sidelengths of a triangle ,

Now,

By expanding the LHS and using Lemma 1 we only have to prove that –

Since they are side-lengths of a triangle , using Ravi’s Substitution ,

the inequality re-arranges to –

where ,

which is true by Lemma 2.

This completes the proof.

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PKH’s solution is just awesome for the problem.