Inequality from Jensen

Let $f, g:\mathbb{R} \rightarrow \mathbb{R}$ be two continuous functions. If $f$ is convex then prove that:

$$f \left( \int_0^1 g(x) \, {\rm d}x \right) \leq \int_0^1 f\left(g(x)\right)\, {\rm d}x$$

Limit and Riemann sums

Evaluate the limit:

$$\lim_{n\rightarrow +\infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$$

Solution