Analysis Integration Need some help working through a proof.  Not quite sure where to go with this one. 
Suppose that $\{c_n\}$ is a nonconstant sequence of real numbers which converges to $0$. Let $a$ be the infimum of all terms in the sequence and let $b$ be the supremum of all terms in the sequence. Define $f \colon [a,b] \to \mathbb R$ by
$$f(x) = \begin{cases}
1 & \text{ if } x=c_n \text{ for some } n \in \mathbb N, \\
0 & \text{ otherwise.} \end{cases}$$
Prove that $f$ is integrable.
 A: Suggestion: your function $f: [a,b] \rightarrow \mathbb{R}$ has the following two properties:
(i) It is bounded.
(ii) For any $\delta > 0$, it has only finitely many discontinuities on $[a,-\delta] \cup [\delta,b]$.  
I claim that any such function is Riemann integrable.  I'm assuming that you know that any bounded function with only finitely many discontinuities is Riemann integrable: if not, you should establish this first in the case of a single discontinuity, and then in general by (e.g.) induction on the number of discontinuities.  Then using partitions which include $-\delta$ and $\delta$ as sample points, you should be able to bound $U(f,\mathcal{P}) - L(f,\mathcal{P})$ above in terms of $\delta$ and the maximum value of $f$.  Then take $\delta$ to $0$ to deduce the result.
A: Apparently, for all partitions of $[a,b]$, the lower Riemann sum is equal to $0$. It thus suffices to show that for every $\varepsilon>0$, there exists a partition 
$P=\{a=t_0<t_1<\cdots<t_n=b\}$ of $[a,b]$, with $U(f,P)<\varepsilon$. As $c_n\to 0$, there are only finitely many terms with $|c_n|\ge \varepsilon/4$. Assume that
$$
|a_{j_1}|,\ldots,|a_{j_k}|\ge\frac{\varepsilon}{4}
$$
and all the rest terms lie in $[-\varepsilon/4,\varepsilon/4]$. The sought partition $P$ should simply include the following points: $-\varepsilon/4,\varepsilon/4$ and 
$$
a_{j_i}-\frac{\varepsilon}{4k},a_{j_i}+\frac{\varepsilon}{4k}, \quad i=1,\ldots,k.
$$
