Suppose a sequence $(X_n)_{n \geq 0}$ of random variables converge in distribution to a random variable $X$, where $X_{n}>0$ and $X\sim $exp$(1)$. Suppose further that $(Y_{n})_{n \geq 0}$ converge in probability to $0$. Define
$K_{n}=\max \left\{\frac {1}{X_{n}},Y_{n}\right\}$
My question is whether the limiting distribution of $K_{n}$ is $\frac {1}{X}$. In other words, is taking the maximum and taking the limit be swapped?
If we show that $(\frac 1 {X_n},Y_n) \to (\frac 1 {X},0)$ in distribution then we are done since $(x,y) \to \max \{x,y\}$ is a continuous function on $\mathbb R^{2}$. One way of proving that $(\frac 1 {X_n},Y_n) \to (\frac 1 {X},0)$ in distribution is to use characteristic functions. We have $E|e^{it\frac 1 {X_n}+isY_n}-e^{it\frac 1 {X_n}}|\leq E|e^{isY_n}-1| \to 0$ by DCT so we are done.