\displaystyle $\begin{align*}
\mathbb{P}(X \geq 2 Y) & = \mathbb{P} \left( \bigcup_{n = 1}^{\infty} (Y = n, X \geq 2 n) \right) = \sum_{n = 1}^{\infty} \mathbb{P}(Y = n) \mathbb{P}(X \geq 2 n) = \sum_{n = 1}^{\infty} \mathbb{P}(Y = n) \mathbb{P} \left( \bigcup_{k = 2 n} X = k \right) \\
& = \sum_{n = 1}^{\infty} \mathbb{P}(Y = n) \sum_{k = 2n}^{\infty} \mathbb{P}(X = k)
\end{align*}$