# Convex subset

Let $A$ be the subset of $\mathbb{R}^2$ given by
$A = \{(x,y) \hspace{1mm} | \hspace{1mm} y\leq x, y\leq 2\}$. Show that A is convex.

Please help me check the steps below which shows that A is indeed convex by using the fact that the intersection of two convex subsets is convex.

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• Imanonymous

Just to confirm, since A is convex, then A is indeed pathwise connected right?

• Imanonymous

May I know how we can get tx_1+(t-1)x_2<=ty_1+(1-t)y_2?

• Nirenberg

You indeed have the inequality in the opposite order and just revised my solution. That because y_1 <=x_1 and y_2 <= x_2.

• Imanonymous

Is it still correct if I rewrite it as y_1-x_1<=0? Just like what I wrote in my proof.

• Nirenberg

Yes, but I prefer the way I have written is as it is closer to the definition of A_1.

• Imanonymous

Alright. Thank you for your help :)

• Nirenberg

Yes.