Calculate the limit of a sequence: $\frac{ a_n(y_n) }{e^{y_n}}$

Question

Let $(a_n)$ be the Taylor development of the exponential function as follows: \begin{eqnarray} a_n(x) = \sum^{n}_{k = 0} \frac{x^k}{k!} \end{eqnarray}

Let $(y_n)$ be a positive and strictly monotone sequence with limit $+\infty$.

Question Can we prove the following limit ?

\begin{eqnarray} \lim_{n \to +\infty } \frac{ a_n(y_n) }{e^{y_n}} = 1 \end{eqnarray}

If not possible, can we prove that at least this limit $\lim_{n \to +\infty } \frac{ a_n(y_n) }{e^{y_n}} > 0$ ?

You can find a French version of this question here: Calculer la limite de la suite suivante: $\frac{ a_n(y_n) }{e^{y_n}}$?