> 
> Suppose that $a\_n$ and $b\_n$ are two real sequences satisfying $a\_nb\_n\to 1$ and $a\_n+b\_n\to 2$. Does it follow that $a\_n$ and $b\_n$ both converge to $1$?
> 
> 
> 

I was working on this problem as one of the exercises from [this online source of problems in analysis](http://etananyag.ttk.elte.hu/FiLeS/downloads/4b_FeherKosToth_MathAnExII.pdf). I have not been able to find a counterexample, but I have deduced some necessary conditions on $a\_n,b\_n$:

1. Eventually $a\_n$ and $b\_n$ have the same sign, and this must be $>0$:

If they do not eventually share the same sign, then $a\_nb\_n$ can't converge to $1$. If they are not both positive, their sum can't converge
to $2$.
2. There exists $t>0$ and $N$ such that $a\_n>t$ for all $n\ge N$. [Hence the same conclusion holds for $b\_n$.]:

If not, there is some subsequence $a\_{n\_k}$ with $a\_{n\_k}\to 0$. But then
$b\_{n\_k}\to\infty$ as $a\_{n\_k}b\_{n\_k}\to1$. Then $a\_{n\_k}+b\_{n\_k}\to\infty$, absurd.
3. $a\_n$ and $b\_n$ are both bounded above:

Otherwise, there must be a subsequence of the other that tends to $0$, which is impossible by 2.

I imagine a counterexample would be very strange, but I feel like a proof is also not too far out of reach.

Thanks for any suggestions.

---

P.S. Does the conclusion change if we allow complex sequences $a\_n,b\_n$?