# Math Help - Proving a noSet is Convex

1. ## Proving a noSet is Convex

Hi there

How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

Thanks

*Sorry for the title typo

2. Originally Posted by Lior539
Hi there

How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

Thanks

*Sorry for the title typo

Take any two points $u:=(x_1,x_2),\,v:=(y_1,y_2)\in S$ ; you must prove that $tu+(1-t)v\in S\,,\,\,\forall\,t\in [0,1]$:

$tu+(1-t)v=\left(tx_1+(1-t)y_1,\,tx_2+(1-t)y_2\right)$ , and now you have the simple task yo show that:

1) $tx_1+(1-t)y_1\geq 0$ ;

2) $tx_1+(1-t)y_1+tx_2++(1-t)y_2\geq 1$ ...

Tonio

3. Thanks for your help Tonio, it really cleared up some things in my mind. How would one go about proving (1) and (2) though?

Thanks