Exercise 11, § 7

Consider a sequence $(a_n)$ every term of which is the sum of a finite sequence of finite real numbers

For each pair $(n, p)$ of positive integers such that $p \leq k_n$, if $m = \sum_{i=0}^{n-1} k_i + p$, put $c_m = b_{n,p}$. Show that if the series whose general term is $a_n$ is convergent in $\mathbf{R}$, and if

tends to 0 as $n$ tends to infinity, then the series whose general term is $c_m$ is convergent and has the same sum as the series whose general term is $a_n$.