Proving that if $c^2 = a^2 + b^2$ then $c < a + b$ I am having some trouble in proving a conjecture that occurred to me some time ago, based on the Pythagorean theorem.
If, for a non-degenerate triangle, $$c^2 = a^2 + b^2$$
Then can the following be proven?
$$c < a + b$$
Is this statement always true?

For all $a,b \in \mathbb{Z}^+$ we have $a^2 + b^2 < (a+b)^2 \implies \sqrt{a^2+b^2} < a+b$. If $c^2$ is defined to be $c^2 = a^2 + b^2$, then we have $c < a+b$. This also explains why the triangle cannot be degenerate, i.e. $a = 0$ or $b = 0$, in order for statement to be true.
