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# proof: $\lim_{n\rightarrow \infty } \int_{0}^{a}f(nx)dx = aL$ when $\lim{x\to\infty }f(x)= L$

we know that:

$$\lim{x\to\infty }f(x)= L$$

$$\alpha > 0$$

i need to prove that:

$$\lim_{n\rightarrow \infty } \int_{0}^{a}f(nx)dx = aL$$

my idea was to get the limit inside the integral and then i will have this:

$$\int_{0}^{a}L = L(a-0)$$

but i am not sure that it is correct