Prove series converge for almost every $x$ 
Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in \mathbb{R}$.

The question came from old qualifying exam.
Although not specified, in the context of the test measure on $\mathbb{R}$ is always Lebesgue measure.
I define $q=(1-\frac{1}{p})^{-1}$. The integral can be estimated by Hölder's inequality $$\begin{align}\int_n^{n+n^{-\alpha}} |f(x+y)|dy&=\int_{n+x}^{n+n^{-\alpha}+x} |f(y)|dy\\&=||f||_1\\&\leq||f||_p||1||_q\\&=\left(\int_{n+x}^{n+n^{-\alpha}+x} |f(y)|^pdy\right)^{\frac{1}{p}}(n^{-\alpha})^{\frac{1}{q}}\end{align}$$
Here $||\cdot||_p$ means the norm on $L^p \left([n+x,n+n^{-\alpha}+x]\right)$.
Suppose we can prove the integral $\int_{n+x}^{n+n^{-\alpha}+x} |f(y)|^pdy$ decays faster than $n^{-(1+\frac{1}{q})}$ then the $n^{th}$ term should decay faster than $\frac{1}{n}$ and therefore converge. However this is the farthest I can get, any idea how to proceed?

Suppose $f\ge0$ to save typing. Let $q$ be the conjugate exponent.
Now
$$\begin{aligned}\int_0^1\int_n^{n+n^{-\alpha}}f(x+y)\,dydx
&=\int_n^{n+n^{-\alpha}}\int_0^{1}f(x+y)\,dxdy
\\&=\int_n^{n+n^{-\alpha}}\int_y^{y+1}f(x)\,dxdy
\\&\le \int_n^{n+n^{-\alpha}}\int_n^{n+2}f(x)\,dxdy
\\&= n^{-\alpha}\int_n^{n+2}f(x)\,dx\end{aligned}.$$
So if $F(x)$ is that sum then $$\int_0^1F(x)\,dx\le\int fK,$$
where $$K=\sum_{n=1}^\infty n^{-\alpha}\chi_{(n,n+2)}.$$
The fact that $\alpha q>1$ shows that $K\in L^q$. So $\int_0^1 F<\infty$, hence $F(x)<\infty$ for almost every $x\in[0,1]$. Similarly for $x\in[k,k+1]$.
