Show that the set $\{(x,y): 0\leq x,y \leq 1, x+y \leq 1\}$ belongs to the $\sigma$-algebra $\mathcal{B}(\mathbb{R}^2)$
I know in general that any compact subset belongs to the Borel $\sigma$-algebra. What could be the proof for, let's say, a triangle?
My thoughts is to use the definition the any rectangles form Borel $\sigma$-algebra by definition. This is, we can make a product of two intervals to get this set. Since $[0,1] = \cap_{n=1}^\infty(0-\frac{1}{n},1]$, $[0,1]$ belongs to $\mathcal{B}(\mathbb{R})$. What about the other dynamic interval that goes from 1 (at the point 0) to 0 (at the point 1)?