$X$ RV $X(S)=s$. Prove $X_n\underset{a.s}{\to}X$.

Question

Let $([0,1],\mathcal{B}_{[0,1]},\lambda)$ probabilty space and let $X_n(s)=\frac{n}{n+1}s+(1-s)^n$ a sequence for RVs for every $s\in[0,1]$. and let $X$ RV $X(S)=s$.
Prove $X_n\underset{a.s}{\to}X$.

I've tried using Borel-Cantelly and I got
$$\sum\mathbb{P}(|X_n-x|\geq\epsilon)=\sum(1-\mathbb{P}(|X_n-s|\leq\epsilon)=\sum(1-\mathbb{P}(|-\frac{s}{n+1}+(1-s)^n|<\epsilon)$$
Couldn't figure how to continue from here.

Answer 1

No theorem is needed. Just the definition of almost sure convergence.

For every $s \in (0,1]$ we have $\lim X_n(s)=(1)(s)+0=s=X(s)$. And $\lambda (\{0\})=0$. This proves that $X_n \to X$ almost surely.