Suppose that f and g are real valued functions defined on some interval (a,b) containing the point $\displaystyle x_0$. Suppose also that $\displaystyle \lim_{x \to x_0}f(x) =A$ and $\displaystyle \lim_{x \to x_0}f(x) =B$. Prove that if $\displaystyle f(x)<g(x)$ for all $\displaystyle x \in (x_0 - \phi,x_0+\phi)$ (for some $\displaystyle \phi >0$). Then $\displaystyle A \leq B$. In this case is it always true that A<B.

For the first part my proof so far consists of a contradiction. I have assumed A>B and then proceded to write out the definitions of limit tending to $\displaystyle x_0$ with $\displaystyle \epsilon, \delta$ and so on. However I can't seem to find a contradiction and was wondering if someone could give me the first few lines I need.

I also have no idea for the second part. Thanks!