On page 3 of W S Anglin's
Mathematics: A Concise History and Philosophy,
there is the statement that
In 1880 J J Sylvester proved that any proper fraction can be expressed
as a sum of distinct unit fractions.
Then Angelin gives a very brief and easy to follow proof by mathematical
induction, which, I suppose is approximately the proof that Sylvester gave.
Angelin doesn't claim that Sylvester's is the first proof, and I would be
surprised to find that it hadn't been proved until 1880. Where by the way
do we find Sylvester's proof?
Best wishes,
Sam Kutler