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**redsoxfan325** Problem:

Here's what I have so far.

I want to try to prove that $\displaystyle \sum_{n=1}^{\infty} f(n)$ converges. Since $\displaystyle f$ is uniformly continuous, $\displaystyle \forall~ \epsilon>0, \exists~ \delta>0$ such that $\displaystyle |f(x_i)-f(x_{i-1})|<\epsilon$ when $\displaystyle |x_i-x_{i-1}|<\delta$.

Let $\displaystyle P$ be a partition of $\displaystyle [0,\infty)$ such that $\displaystyle \Delta x_i < \delta$. Let $\displaystyle m_i = \inf f(x)$ on $\displaystyle [x_{i-1},x_i]$ and let $\displaystyle M_i = \sup f(x)$ on $\displaystyle [x_{i-1},x_i]$. We know that:

$\displaystyle 0 < \sum_{i=1}^{\infty} m_i\Delta x_i < \sum_{i=1}^{\infty} m_i\delta $ $\displaystyle \leq \int_0^{\infty}f(x)\,dx \leq \sum_{i=1}^{\infty} M_i\Delta x_i$ $\displaystyle < \sum_{i=1}^{\infty} M_i\delta$

Now I get stuck. I was thinking that since I know $\displaystyle |f(x_i)-f(x_{i-1})|<\epsilon$, I could put some restrictions on $\displaystyle m_i$ and $\displaystyle M_i$, but I'm not sure how to do this. Am I on the right track? Regardless, can someone give me a hint as to the best way to proceed?