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The Marlboro Man, c. 17; used for cigarette advertising.Q:
Proof of equality of the sequences $a_n$ and $b_n$
Suppose that $\{a_n\}$ and $\{b_n\}$ are two sequences of real numbers which satisfy the following:
$
\begin{cases}
b_{n+1}-a_n=2a_n+1\\
a_{n+1}-b_n=2b_n-1
\end{cases}
$
Prove that $\forall n\in\mathbb N\ \ a_n=b_n$.
How would you do this? My idea was the following: since the sequence $\{b_n\}$ is increasing, then it must have a limit and the sequence $\{a_n\}$ must also have a limit. But I can't think of a way to prove that the two sequences converge to the same value.
A:
Hint: $\;\,a_n=b_n-1\iff a_n=b_n-2$.
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