Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence $$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$ is convergent.
Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence
$$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$
is convergent.