Research papers in computer science often contain both positive and negative results. A positive result is, usually, an algorithm that solves a certain problem. A negative result is a proof that an algorithm does not exist.
I am looking for a phrase to use when moving from the presentation of a negative to the presentation of a positive result, or vice versa.
The above theorem showed that an exact solution to problem X does not exist.
[requested-phrase-here], an approximate solution does exist, as we show now.
We have just proved a lower bound on the run time of any algorithm solving problem Y.
[requested-phrase-here], we now prove an upper bound.
I can use "However", but I want a phrase that emphasizes that we move from the negative part of the paper to the positive part. Maybe something like: "On the positive side", or: "The good news are that...". What do you suggest?