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.


Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
  • Just to confirm, since A is convex, then A is indeed pathwise connected right?

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

  • 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.

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

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

  • Alright. Thank you for your help :)

  • Yes.

The answer is accepted.