Prove that $\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty$, where $|r_1|, |r_2| < 1$

Question

I'm trying to prove that:

$$ \sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty $$

where $|r_1|, |r_2| < 1$ and $c_1, c_2$ are some arbitraty real numbers ($r_1, r_2$ are also real numbers). If we know that e.g. $r_1, r_2, c_1, c_2$ are all positive, then this is easy because then we have two infinite geometric series, which are know to converge given $|r_1|, |r_2| < 1$. But if e.g. $r_2<0$ the summands in the series don't decrease in a geometric fashion and we must find another way of making sure that the sum of the series is finte. How would one do that?

Answer 1

HINT

We have that by triangle inequality

$$|c_1r_1^j + c_2r_2^j| \le |c_1r_1^j| + |c_2r_2^j|$$

Answer 2

$$\sum_{j=0}^\infty |c_1r_1^j + c_2r_2^j| \le \sum_{j=0}^\infty |c_1||r_1|^j + \sum_{j=0}^\infty|c_2||r_2|^j$$

By comparison test, it converges.