Putnam 2017 A3

Let $a$ and $b$ be real numbers with $a<b$, and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0, \infty)$ such that $\int_a^b f(x) dx = \int_a^b g(x) dx$ but $f \neq g$. For every positive integer $n$, define $$I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n} dx.$$ Show that $I_1, I_2, I_3, \dots$ is an increasing sequence with $\lim_{n \to \infty} I_n = \infty$.