Spring 2018, problem 68

Let $f, g\colon [0,1] \rightarrow (0,+\infty)$ be non-equal, continuous functions such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$. Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq \mathbf{N}$. Prove that $(y_n)$ is an increasing and divergent sequence.