Proving that if $a^2+b^2=c^2$, then $a+b\ge c$.

real-analysis

Hello, I'm trying to prove this statement.

Let a,b & c be three positive real numbers and if $a^2+b^2=c^2$ then $a+b\ge c$

Any help, please?


$(a+b)^2=a^2+b^2+2ab\geq a^2+b^2= c^2\rightarrow (a+b)^2\geq c^2\rightarrow a+b\geq c$


Hints:

(1) If for positive real numbers $\;a,b,c\;$ we have that $\;a^2+b^2=c^2\;$ , then there exists a right triangle with legs $\;a,b\;$ and hypotenuse $\;c\;$

(2) In Euclidean Geometry : the sum of the lengths of any two sides of any triangle is greater than the length of the third side.

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