# Prove or disprove: Justify your answer using the def or limit. Real analysis

1. Apr 6, 2012

### Hodgey8806

1. The problem statement, all variables and given/known data
Suppose f: ℝ-{0} → ℝ has a positive limit L at zero. Then there exists m>0 such that if 0<|x|<m, then f(x)>0.

2. Relevant equations
The definition of the limit of a function at a point is: (already assuming f to be a function and c being a cluster point)
A real number L is said to be a limit of f at c if, given any ε>0, there exists a δ>0 such that if x is an element of the domain and 0<|x-c|< δ, then |f(x)-L|<ε.

3. The attempt at a solution
I'm proving this true as:
Spse the limit as x→0 of f = L (being a positive real number.
This implies that given ε>0, there exists δ>0 such that 0<|x-0|< δ, then |f(x) - L|<ε.
Let ε=L,
By definition of the limit of a function at a point there exiss δ>0 such that for x in ℝ - {0}, 0<|x|<δ and |f(x) - L|<L if and only if 0<f(x)<2L.
Thus taking m = δ, then for 0<|x|<m and f(x)>0.

I'm sure it is sloppy, but it does make sense to me here. Please, constructive criticism is welcome! :)

2. Apr 6, 2012

### Dick

Makes sense to me.

3. Apr 6, 2012

### Poopsilon

I agree this is actually quite well done.

4. Apr 6, 2012

### Hodgey8806

Thank you very much! Finally analysis is coming back around to itself so I can sort of see where things are going easier :)