First argument: people will be willing to use it. Consider what you're offering -- a closed-source library that manages people's financial transactions. You may or may not be surprised to hear that people are reluctant to trust closed-source libraries that process financial transactions, account details, and patterns of economic behavior.

Second argument: open-source libraries are more likely to be successfully scrutinized for security flaws, and repaired in advance of exploitation. An irony of open-source is that over time it becomes more secure, not less, indeed one often-heard explanation for Windows' notorious insecurity is that it is closed-source, so the first revelation of a vulnerability is, not when errors are noticed by routine scans of source code, but when they're exploited in the wild.

Third argument: people will be able to see how your code works. This isn't a license to steal, it's a way for people to find out if your code meets their needs and behaves as they expect (or as accounting standards require).

I once debugged an accounting package that had a weird error -- two runs on the same transaction database never produced the same outcome. After a lot of back and forth, I got the developer to open the source, and discovered that his way of rounding off to the nearest penny was to generate a random binary number -- if 0, round down, if 1, round up. It was breathtakingly stupid, and we would never have solved it without seeing the source.

Fourth argument -- you can't arbitrarily shut down users of your library. An open-source library survives the extinction of its source or a change in policy. A closed-source library can't do that in any meaningful sense.

That should do. :)