Prove:if $b_{n}\rightarrow 0$ then $\sum_{n=m}^{\infty} a_{n}=b_{m+k}+b_{m+k+1}+b_{m+k+2}+...+b_{m+k+l-1}$ Suppose that $m,k ∈ {\Bbb N_{0}}$, that $l ∈{\Bbb N}$, that $(a_{n})_{n=m}^{\infty}$ and $(b_{n})_{n=m}^{\infty}$  are sequences of real numbers, and that $a_{n} = b_{n+k} −b_{n+k+l}$ for all $n ≥ m$ where $n ∈ {\Bbb N_{0}}$ 
Prove that:
if $b_{n}\rightarrow 0$ then
$\sum_{n=m}^{\infty} a_{n}=b_{m+k}+b_{m+k+1}+b_{m+k+2}+...+b_{m+k+l-1}$
But when I try to prove it. Simply I have got:
$\sum_{n=m}^{\infty} a_{n}$
$=\sum_{n=m}^{\infty}(b_{n+k} −b_{n+k+l})$
$=\sum_{n=m}^{\infty}b_{n+k} −\sum_{n=m}^{\infty}b_{n+k+l}$
$=\sum_{n=m+k}^{\infty}b_{n} −\sum_{n=m+k+l}^{\infty}b_{n}$
$=\sum_{n=m+k}^{m+k+l-1}b_{n}+\sum_{n=m+k+l}^{\infty}b_{n} -\sum_{n=m+k+l}^{\infty}b_{n}$
$=\sum_{n=m+k}^{m+k+l-1}b_{n}$
It seems that I didn't use the fact that $b_{n}\rightarrow 0$.
So I guess I've got something wrong.
Could someone tell me how to prove that exactly? Where did I make the mistake? How can I use the fact that $b_{n}\rightarrow 0$?
Thank in advance!
 A: You assumed convergence of the series $\sum\limits_{n = m}^\infty b_{n+k}$, which was not a given. To use the fact that $\lim\limits_{n \to \infty} b_n = 0$, first consider the partial sums of the series $\sum\limits_{n = m}^\infty a_n$:
$$\sum_{n = m}^N a_n = \sum_{n = m}^N (b_{n+k} - b_{n+k+l}) = \sum_{n = m}^N (b_{n+k} - b_{n+k+1}) + \cdots + \sum_{n = m}^N (b_{n+k+l-1} - b_{n+k+l})\tag{*}\label{eq1}$$
Each summand on the right hand side of \eqref{eq1} is a telescoping sum, with values $b_{m+k} - b_{N+k+1},\ldots b_{m+k+l-1} - b_{N+k+l}$, respectively. Thus 
\begin{align}\sum_{n = m}^N a_n &= (b_{m + k} - b_{N+k+1}) + \cdots + (b_{m+k+l-1} - b_{N+k+l})\\
&= (b_{m+k} + \cdots + b_{m + k + l-1}) - (b_{N + k + 1} + \cdots + b_{N+k+l})\tag{**}\label{eq2}\end{align}
In the last line of \eqref{eq2}, the second group of terms has $l$ terms, each tending to $0$ as $N\to \infty$. Thus $b_{N+k+1} + \cdots + b_{N+k+l} \to 0$ as $N\to \infty$; consequently,
$$\sum_{n = m}^\infty a_n = b_{m+k} + \cdots + b_{m+k+l-1}$$
