
question on a proof
I need a little confirmation on the following proof:
The problem is this:
For an interval I, with f:I > R where f is strictly increasing with f(I) an interval, prove that f^(1):R > I is strictly increasing.
My proof is this:
Because f is strictly increasing, for y,z exists in I such that y<z,
f(y)<f(z).
Let y,z exist in I. Then f(y),f(z) exist in R. and f(y) < f(z).
but f^(1)[f(y)]<f^(1)[f(z)] which implies y<z and thus f^(1) is strictly increasing.
This proof seems trivial to me, which makes me think I did something wrong. I possibly assumed too much in my proof, not sure. Any ideas?

Hello
I suffer from similar habbits. Always assuming things are complicated. Your proof is correct in its simplicity. You are absolutely right to assume the existence of $\displaystyle f^{1}$, its given in the problem statement. Hence you may use the identity function $\displaystyle f^{1}(f(x)) = x$, by the definition of the inverse.
In fact just to reassure you that your proof makes sense, remember that f is the inverse of the function $\displaystyle f^{1}$. And a function only has an inverse if it is strictly increasing or decreasing (a standard theorem from set theory). You have of course proved $\displaystyle f^{1} $ is strictly increasing. And thus the inverse of $\displaystyle f^{1}$ exists. namely, f.