Proving that if the odd and even sequences converge, where one is decreasing and the other is increasing, then they converge to the same limit

Question

I'm trying to prove this statement for this sequence as a part of a larger proof. The specific sequence I'm looking at is : $b_{n+1} = 1+\dfrac{1}{b_n}$, where $b_1 = 2$. I'm attempting to show that ($b_{2n}) \rightarrow A \land (b_{2n-1}) \rightarrow B \implies A=B$. I know that this statement is not true in all cases, e.g. if $b_n = (-1)^n$. For this reason, I think I might have to use the fact that for this specific sequenence, $(b_{2n})$ is monotonic increasing and $(b_{2n-1})$ is monotonic decreasing. Note that I am not assuming that $(b_n)$ converges from the start, and I am not trying to show that it converges to the same value as the subsequences. Is the statement: "if the odd and even sequences converge, where one is decreasing and the other is increasing, then they converge to the same limit" even true in general? If so, how do I go about proving it? I'm completely stuck on how to proceed with the information that I seem to have.