I am trying to prove a result I need in a proof, which is the following. Say I have two real values a,b and I can get the one as close to the other as I want |a-b|<ε then I want to prove I can get 1/a 1/b as close as Ι want to the other. |1/a - 1/b|<ε'.
a=a(y),b is a constant (it's part of a limit proof for the inverse function derivative)
I've proven so far |1/a - 1/b|=|(a-b)/ab| < ε/|ab| Now as ε gets smaller ab approaches a constant value b^2 so I can make |1/a - 1/b| as small as I want. How do I make the last argument rigorous?