Let $X_n$ be a sequence of i.i.d. random variables (rv), $X$ a rv and $a_n$ a positive sequence of numbers decreasing to zero such that there is $c>0$ for which $$P(|X_n-X|\geq c a_n)\to 0, \text{ as } n\to\infty,\quad (1)$$ in asymptotic notation, $X_n=X+O_p(a_n)$. How can I use this information to find the rate at which $e^{X_n-X}-1$ is bounded in probability? In other words, for which sequence $b_n$ there is $c_2>0$ such that $$P(|e^{X_n-X}-1|\geq c_2b_n)\to 0, \text{ as } n\to\infty? \quad (2)$$
Since $(1)$ holds only for a specific $c>0$, I believe I can't use the continuous mapping theorem to conclude anything about $(2)$. Perhaps, I need to use some exponential inequality.
Can you help me with this?
Thank you in advance.