Given two strictly increasing bounded sequences of reals $x_n$ and $y_n$. What is known about existence of real analytic function $f$ with property $f(x_{n_k})=y_{n_k}$ for some subsequence $x_{n_k}$ ?

**Edit 1:** Of course, $y_n$ can't increase too slow or too fast relative to $x_n$ as it was shown in counterexamples below. So, we can additionally assume the following ($x=\sup x_n,~y=\sup y_n$):

$0<\underline\lim\limits_{n\to\infty}\frac{|x_n-x|}{|y_n-y|},~ \overline\lim\limits_{n\to\infty}\frac{|x_n-x|}{|y_n-y|}<\infty$

or even: $0<\lim\limits_{n\to\infty}\frac{|x_n-x|}{|y_n-y|}<\infty$ if needed.

**Edit 2:** Clear counterexample is given below. Can one suggest any sufficient conditions on $x_n, y_n$ for existence of $f$ ?