$\{X_n\}$ and $\{X\}$ are random variables defined on $(S,σ,P)$ with $E(X_n^2)<\infty$, for all $n=1,2,3,\dots$, and $E(X)<\infty$. If $E\left((X-X_n)^2\right)<\infty$ as $n\to\infty$, then I'd like to show that the sequence $\{X_n\}$ converges to $\{X\}$ in probability as $n\to\infty$.
My teacher asked me to use convergence in moments however I am not able to make any progress in this problem.
There is a typo in your post.
If
$$\lim\limits_{n\to\infty}\mathbb{E}[X_n-X]^2=0$$
That is
$$X_n\xrightarrow{L^2}X$$
than it is also
$$X_n\xrightarrow{\mathcal{P}}X$$
Proof:
Using Chebishev inequality we get
$$\lim\limits_{n\to\infty}\{|X_n-X|\leq \epsilon\}\geq 1-\frac{\mathbb{E}[X_n-X]^2}{\epsilon^2}$$