Could someone please check my work

pup Please state all definitions and theorems that you will need: Definition 5.1.1 Let f: D - R and letc be an accumulation point of D. We say that a real number L is a limit of fat c , if for each E > 0 there exists a o > 0 such that If(x) - L| 0. Prove that there exists a deleted neighborhood U of c such that f(x) > 0 for all x E Un D. Suppose f: D - R and let c be an accumulation point of D . Suppose lim f(x) > 0. Since lim f(x) > 0 , then it is given thatL > 0 . Let & L then by definition 5.1.1 for the limit of a function, since L is a limit of f at c , then for each & > 0 38 > 0 such that x E D and 0 L then f(x) > 0 when & E U n D since L is a positive number. If f(a) 0. Therefore, if f: D -+ R and c is an accumulation point of D and lim f(x) > 0 , then there exists a deleted neighborhood U of c such that f(x) > 0 for all & E Un D