Could someone please check my work

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 f at c , if for each & > 0 there exists a o > 0 such that If (x) - L| 0 38 = - (choosing o to be infinitesimally small) such that 3 8n E D such that 0 E, which means (f(Sn)) will never be within & of a limit, L , which means (f(Sn)) is not convergent to L . Since (f(Sn)) is not convergent to L , the sequence (f(Sn)) is not convergent in R. Therefore, if f does not have a limit at c then there exists a sequence (Sn) in D with each In # c such that (Sn) converges toc, but (f(Sn) ) is not convergent in R. 2. (b) = (a) If there exists a sequence (Sn) in D with each Sn # c such that (Sn) converges to c , but (f(Sn)) is not convergent in R then f does not have a limit at c. Suppose there exists a sequence (Sn) in D with each Sn # c such that (Sn ) converges to c and (f(Sn)) is not convergent in R. Now suppose by contradiction that f has a limit at c . Then by definition 5.1.1, for each & > 0 36 > 0 such that 38n E D such that 0