If $f_n \to f$ uniformly then $\int_{0}^{1-\frac{1}{n}}f_n \to \int_{0}^{1}f$.
I have two proofs and I was wondering if both of them are correct and if there are any other neat solutions 1.
Consider $F_n(x)=\int_{0}^{x}f_n$, then $F_n(x) \to F$ uniformly by uniform convergence of $f_n$. Thus $F_n(1-\frac{1}{n}) \to F(1)$ and we are done
2. Since $f_n$ converges, it is a bounded sequence $|f_n|\leq M$ thus we can apply DCT on $|\chi_{[0,1-\frac{1}{n}]}f_n|\leq M$ and so the limit is $\int_{0}^{1}f$.