I'm having trouble proving this statement:
Prove that the set
$$X=\{ (x,y)\in\mathbb{R}^2 : y\geq 0, x\leq b+y \}$$
given $b\in\mathbb{R}$ is convex.
The work I've done so far:
I first assumed we havo two different points $(x_1,y_1),(x_2,y_2)\in X$. Then, the segment between them is made of the points
$$(\lambda x_1 + (1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2) \ , \ \lambda\in[0,1].$$
So if I prove that points are inside $X$ for any $\lambda$ I would have proven the convexity.
It's easy to see that
$$\lambda y_1 + (1-\lambda)y_2 \geq 0,$$
since it's a positive number plus another positive number. Now, I don't know how to prove that
$$\lambda x_1 + (1-\lambda)x_2 \leq b + \lambda y_1 + (1-\lambda)y_2$$
knowing obviously that $x_1 \leq b+y_1$ and $x_2\leq b+y_2$.
What can I do to prove this part? Any help or hint will be appreciated, thanks in advance.
Best Answer
Just multiply $x_1 \leq b+y_1$ by $\lambda$, $x_2 \leq b+y_2$ by $(1-\lambda)$ and add the inequalities. Finally note that $\lambda b +(1-\lambda) b=b$.