"further convincing" is correct and usable in written English.
You would use this phrase when you are trying to present additional evidence or facts that support your argument or opinion. For example, "In addition to the statistical evidence, further convincing arguments can be found in the testimony of key witnesses."
But, for those of you who need further convincing that we can do away with conference calling, allow me to explain.
Over successive rounds, each side could bring less to the table or be less willing to follow through on its promises — further convincing one another that talks are not worthwhile.
We can talk about it constantly through the entire next administration, further convincing all Americans under the age of 50 that the older generation is never going to get out of the way and let them have the good jobs.
Charles Davis, an economist at the Centre for Economics and Business Research, added: "As the deficit rises, chancellor Darling may well need to find a way of further convincing the markets of the UK government's credibility - with the possibility of further future spending cuts and/or tax rises.
BBC Persian increased its time on air by five hours a day to 13 for four days after the election, further convincing some that it is a tool of the British government.Iran's best-loved television serial, "My Uncle Napoleon", pokes fun at a man who sees the stealthy hand of British imperialism behind every event in Iranian history.
We also found this exact same result in a laboratory study (Koskella and Lively 2007; Koskella and Lively 2009), further convincing us that parasites, rather than correlated environmental factors, are driving the observed changes.
Many Palestinians will be further convinced by the new report that Israel was behind his death.
This will only further convince diehards that he is a traitor to conservatism.
Obama appeared to further convince Xi on the need to rein in Pyongyang.
Recently I have been further convinced by going for long walks with my children and our new puppy.
Additional visible light observations further convinced the scientists that they were not being fooled by variations in the star itself erroneously mimicking the presence of a planet.
