# Several ways to prove Hardy's inequality

Desvl at
Suppose $$1 0$$ only if $$0 0$$, there exists some $$m$$ such that $$\lrVert[f_n-f]_p < \frac{1}{n}$$. Thus \[\begin{aligned}|F_n(x)-F(x)| &= \frac{1}{x}\left\vert \int_0^x f_n(t)dt - \int_0^x f(t)dt \right\vert \\ &\leq \frac{1}{x} \int_0^x |f_n(t)-f(t)|dt \\ &\leq \frac{1……