Show convergence of $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} c_n$ implies the convergence of $\sum_{n=0}^{\infty} b_n$

Question

Let $(a_n), (b_n), (c_n)$ be sequences of real numbers such that $\forall n \in \mathbb{N} \ \lvert a_n-b_n \rvert \leq c_n$. I want to show that if the series $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} c_n$ converge, the series $\sum_{n=0}^{\infty} b_n$ also converges. Do you have any hints?

Answer 1

Hint: you can use the completeness of real numbers. For $p\geq q$: \begin{aligned} \left|\sum_{n=0}^pb_n-\sum_{n=0}^qb_n\right|&=\left|\sum_{n=q+1}^pb_n\right|\\ &=\left|\sum_{n=q+1}^pa_n+\sum_{n=q+1}^p(a_n-b_n)\right|\\ &\leq \left|\sum_{n=q+1}^pa_n\right|+\sum_{n=q+1}^pc_n\\ &\leq \left|\sum_{n=q+1}^pa_n\right|+\left|\sum_{n=q+1}^pc_n\right|. \end{aligned} You can make the two expressions on the last line arbitrarily small by the Cauchy properties of the sequence of partial sums of $a_n$ and $c_n$, respectively.