sequences and series - If $a_1 + 2a_2, a_2 + 2a_3, a_3 + 2a_4 dots$ converges, prove $a_1, a_2, dots$ converges

Let $a_1, a_2, \dots$ be a sequence of real numbers such that the
sequence $a_1 + 2a_2, a_2 + 2a_3, a_3 + 2a_4 \dots$ converges. Prove
that the sequence $a_1, a_2, \dots$ must also converge.

I tried rewriting the $a$ sequence in terms of the first convergent sequence. Letting

$$b_1 = a_1 + 2a_2, \ b_2 = a_2 + 2a_3, \ b_3 = a_3 + 2a_4, \dots$$ we have $$a_2 = \frac{b_1 - a_1}{2},$$
$$a_3 = \frac{b_2 - a_2}{2} = \frac{2b_2 - b_1 + a_1}{4},$$
$$a_4 = \frac{b_3 - a_3}{2} = \frac{4b_3 - 2b_2 + b_1 - a_1}{8}, \\ \dots$$

I have no experience in analysis so I tried using only the definition of convergence and $\lim_{n \to \infty}a_n$. My definition of convergence is for any $\epsilon > 0$ there exists $N$ such that $|b_n - c| < \epsilon$ for $n \ge N$. However even though I can get a magnitude bound on the alternating sum of $b_1, b_2, \dots$, I'm still unsure of how to deal with the finitely many $b_1, \dots, b_{N-1}$ that aren't within $\epsilon$ of $c$. I think Cauchy sequences might help but I have no experience with them either.