Upper-bound for the probability of continuous random variables

Question

Random variables $X$ and $Y$ are continuous and $X,Y\in[0,\infty)$. Can the probability $\Pr\left(Y < 1+\frac{1}{X}\right)$ be upper-bounded as follows? $$\Pr\left(Y < 1+\frac{1}{X}\right)\le\Pr(Y<2)+\Pr\left(Y<\frac{2}{X}\right)$$ Is there any way to prove it?

Answer 1

Yes. $\{Y<1+\frac 1 X\} \subset \{X\geq 1,Y<1+\frac 1 X\} \cup \{X< 1,Y<1+\frac 1 X\}\subset \{Y<2\}\cup \{Y<\frac 2 X\}$. Note however that you must take $X>0$ so that $\frac 1 X$ is defined.