The problem:
Let $\xi,\eta$ be two independent integrable r.v. such that $\mathsf P\{\xi> 0\} = 1$ and $\mathsf P\{\eta\geq 0\} = 1$ and $ a = \mathsf E[\xi-\eta]>0. $ Check if $ \lim\limits_{k\to\infty}\mathsf P\{(\xi-\eta)+k\xi\geq y\} = 1 $ for any fixed $y$.
I can neither prove that the limit is $1$ nor find a counterexample, so any help is appreciated.
What I've tried so far: since $a>0$ then $p = \mathsf P\{\xi-\eta\geq a\}>0$. Then for a fixed $y$ we have: $ \mathsf P\{(\xi-\eta)+k\xi\geq y\} \geq p\cdot \mathsf P\{a+k\xi\geq y|\xi-\eta\geq a\} $ but even if $\mathsf P\{a+k\xi\geq y|\xi-\eta\geq a\}\to1$ with $k\to\infty$ it woulnd't be sufficient for the original problem.