Prove the following series converges Let $u, v, w$ be real numbers such that $u+v+w=0$.
Suppose that $\{b_k\colon k=0,1,2,\dots\}$ is a sequence of real numbers such that $\lim_{k\to\infty} b_k=0$.
For $k=0,1,2,\dots$ define
$a_{3k}=u b_k,$ $a_{3k+1}= v b_k,$  $a_{3k+2} = w b_k.$
Prove that the series $ \sum_{n=0}^{\infty} a_n$ converges.
Im having a thouhg problem with this one, any idea I will appreciate. 

Let us have $c_{3k}=u,\ c_{3k+1}=v,\ c_{3k+2}=w$.  By the Dirichlet test, we need to show that $\sum_{k=0}^N c_k$ is bounded and that $\lim_{n\to\infty}b_n=0$.
Clearly, we have
$$\sum_{k=0}^N c_k\le\sum_{k=0}^Mc_k$$
for some $M\in[0,1,2]$ and we already have $\lim_{n\to\infty}b_n=0$, meaning that
$$\sum_{k=0}^\infty c_kb_k$$
converges.
