How do you do an epsilon-delta proof?

Answer 1

LOL, all proof is done the same ---- direct, contradiction or induction.

An epsilon-delta proof is what a very conventional definition of limits of functions.

The proper definition of a limit of a function is, given a real(or complex) function f, we say lim f(x) = b as x approaches a if:

For all ε > 0, there exist δ > 0 such that for all x in the domain of f, if x != a and |x - a| < δ implies |f(x) - b|< ε

The intuition is quite easy, though the difficult look. It simply says no matter how small of an interval centered at x = a, we can always squish an interval around f(x) = b into it. Doesn't that mean if x goes to a, where the interval is so small, the point it goes to is b?

Just do everything you can to reduce it to the definition or equivalent.

In Non-standard analysis, there is a slightly different way of doing it.