Let $f\in L^1(\mathbb{R})\cap L^p(\mathbb{R})$ and show that $F(x)=\int_{\infty}^x f(t)dt$ is Hölder continuous with exponent $\frac{1}{p'}$.
So I know I have to show $\exists C>0$ s.t. $\forall x, y\in \mathbb{R}$ we have $|F(x)-F(y)|<C|x-y|^{\frac{1}{p'}}$. What I've got so far is if we let $x>y$ WLOG we should have that $|F(x)-F(y)|=|\int_{\infty}^x f(t) dt-\int_{\infty}^y f(t)dt|=|\int_{y}^x f(t)dt$. I don't know what to do from there but obviously see that I need to involve the fact that $f\in L^1(\mathbb{R})\cap L^p(\mathbb{R})$ but don't see how to use this to get where we need to go.