Suppose $\displaystyle f: [0,\infty) \longrightarrow \mathbb{R}$ is continuous and let $\displaystyle a_n=\int_0^1 f(x+n) \ dx, \ \ n \in \mathbb{N}.$ We know that $\displaystyle \lim_{n\to\infty} a_n=L.$ Evaluate $\displaystyle \lim_{n\to\infty} \int_0^1 f(nx) \ dx.$
2. First: $\displaystyle \int_0^1 {f\left( {nx} \right)dx} = \tfrac{1} {n} \cdot \int_0^n {f\left( x \right)dx}$
Now: $\displaystyle \int_0^n {f\left( x \right)dx} = \sum\limits_{k = 0}^{n - 1} {\int_0^1 {f\left( {x + k} \right)dx} } = \sum\limits_{k = 0}^{n - 1} {a_k }$ thus we have: $\displaystyle \int_0^1 {f\left( {nx} \right)dx} = \left(\tfrac{1} {n} \cdot \sum\limits_{k = 0}^{n - 1} {a_k }\right) \to L$ - remember Cesàro Mean -