Let $f:[0,1]\rightarrow R$ integrable, and $lim_{t\rightarrow 1^{-}} lim_{N\rightarrow\infty}\sum_{n=1}^{N}f(t^{n})(t^{n}-t^{n+1})=A$ for real number A.
Prove that $\int_{0}^{1}f(x)dx = A$
MY ATTEMPT TO SOLVE THIS
I tried to use the Riemann definition for integrals: I took a partition of $[0,1]$ by taking the parts as $t^{i}-t^{i+1}=\Delta t_{i}$ for $i=1,...n$ and than the sum $(\sum_{n=1}^{N}f(t^{n})(t^{n}-t^{n-1})$ would be a Riemann sum, so when $N\rightarrow \infty$ this equals to $\int_{0}^{1}f(x)dx$ by definition.
The only problem is that I don't know how to formalize this solution. Can you help me with this?
Thanks for the help.