If $|X| \leq K$ and $|Y| \leq K$, then how can I show that $|X-Y| \leq 2K$? It seems very obvious intuitively but I am not able to use the triangle inequality here. Does anyone have any ideas? thanks!
First observe that $|x-y| \leq |x| + |y|$ since $|x-y| = |x + (-y)| \leq |x| + |(-y)| = |x| + |y|$ according to the triangle inequality.
Therefore, if we suppose that for arbitrary real numbers $x, y,$ and $k$ the inequalities $|x| \leq k$ and $|y| \leq k$ are true, then
$$|x-y| \leq |x| + |y| \leq 2\cdot k \implies |x-y| \leq 2 \cdot k$$
$$|X - Y| \le |X| + |Y| \le K + K = 2K$$