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[Community Question] Calculus: Definite integral with interval depends on $n$!

One of our user asked:

The following that
$$\int_0^1 f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}f(\frac{k}{n})\frac{1}{n} \, $$
is well-known fact! But if $$\lim_{n\rightarrow\infty}\frac{r_n}{n}=\alpha,$$
then is it true that $$\int_0^{\alpha} f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{r_n-1}f(\frac{k}{n})\frac{1}{n} \, $$?
If this is true, why? Help me with big mercy!!


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