If $\lim_{n\to\infty}\int_0^1f(x+n)dx=2$, prove that $\lim_{n\to\infty}\int_0^1f(nx)dx=2$.

It is given $f:[0,\infty)\to\mathbb{R}$ is a continuous function and $\lim_{n\to \infty}\int_{0}^{1} f(x+n)dx=2$ then prove that $\lim_{n\to \infty}\int_{0}^{1} f(nx)dx=2$.

I can prove that $\lim_{x\to \infty} f(x)=2$ using the mean value theorem.

But then i used M.V.T. in second integral and getting $\lim_{n\to \infty}\int_{0}^{n} \frac{f(u)}{n}du=f(c)$ for some $c\in (0,n)$, but i cant conclude that $c\to\infty$ and $f(c)\to 2$.

I have thought of using Lebesgue dominated convergence theorem and I have proven it but is there any other way to prove this with only elementary real analysis?