Limit of Expectation values involving exponential i.i.d random variables

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent and identically distributed (i.i.d) random variables with $\mathbb{E}[X_1] = 1$ and $\mathbb{V}[X_1] = 1$. Show that$$\lim_{n \rightarrow \infty} \mathbb{E} \Bigl[ \exp \Bigl(-\frac{1}{2n} \bigl(\sum^{n}_{k=1} (X_k-1) \bigr)^2 \Bigr) \Bigr] = \frac{1}{\sqrt{2}}.$$I am stuck on this problem not knowing how to approach it. Is there a hidden application of some theorem or is the proof elementary?
I would appreciate any tips on how to tackle this limit.

Edit: Based on Oscar's answer I wish to write down my attempt.
Let $\mu = \mathbb{E}[X_1] = 1$ and $\sigma^2 = \mathbb{V}[X_1] = 1>0$, then we may examine the exponent and observe that$$-\frac{1}{2n} \Bigl(\sum^{n}_{k=1} (X_k-1) \Bigr)^2 = -\frac{1}{2} \Bigl( \frac{\sum^{n}_{k=1}X_k-n}{\sqrt{n}} \Bigr)^2 = -\frac{1}{2} \Bigl( \frac{\sum^{n}_{k=1}X_k-n \mu}{\sqrt{n} \sigma} \Bigr)^2 =: -\frac{1}{2} Y_n^2$$where we define the standardized variable $Y_n = \frac{\sum^{n}_{k=1}X_k-n \mu}{\sqrt{n} \sigma}$ for $n \in \mathbb{N}$.
The Central Limit Theorem is applicable and yields $Y_n \rightarrow Y \sim \mathcal{N}(0,1)$ in distribution, i.e. $F_{Y_n}(t) \rightarrow F_Y(t) \, \, (n \rightarrow \infty)$ for all $t \in \mathbb{R}$.
Now, we calculate the expectation value:$$\mathbb{E} \Bigl[ \exp \Bigl(-\frac{1}{2} Y^2 \Bigr) \Bigr] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \exp \Bigl(-\frac{1}{2} y^2 \Bigr) p(y) dy = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \exp \Bigl(-\frac{1}{2} y^2 \Bigr) \exp \Bigl(-\frac{1}{2} y^2 \Bigr) dy = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \exp \Bigl(-y^2 \Bigr) dy = \frac{1}{\sqrt{2\pi}} \sqrt{\pi} = \frac{1}{\sqrt{2}}$$

In order to finish the proof we need the statement:$$\lim_{n \rightarrow \infty} \mathbb{E} \Bigl[ \exp \Bigl(-\frac{1}{2} Y^2_n \Bigr) \Bigr] = \mathbb{E} \Bigl[ \exp \Bigl(-\frac{1}{2} Y^2 \Bigr) \Bigr].$$

Questions: Does this hold true?
And in general: If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a Borel-measurable function, when does$$\lim_{n \rightarrow \infty} \mathbb{E} \Bigl[ f(Y_n) \Bigr] = \mathbb{E} \Bigl[ f(Y) \Bigr]$$hold. What conditions must $f$ satisfy?