I've been working through The Four Pillars of Geometry by John Stillwell. In exercise 2.5.3 he asks,
How can we be sure that lengths $a,b,c>0$ with $a^2+b^2=c^2$ actually fit together to make a triangle? (Hint: Show that $a+b>c$)
Lemma: $a+b>c$. Suppose not that $a+b\leq c$ then $a^2+b^2+2ab\leq c^2$ and $2ab\leq 0$ $\Rightarrow\Leftarrow$ a contradiction is reached since $a,b>0$.
So my two questions are, (1) is my proof of the hint correct, and (2) how does that hint lead to the statement that $a,b$ and $c$ do in fact form a triangle.