(a) Let A be the subset of R defined by A := {1-" : n e N}. Find all cluster points of A and justify your answer. (b) Let S C R and let c e R be a cluster point of S. Prove that for each e > 0 the set...

0 the set W := Sn (c- e, c+ e) has infinitely many elements. Let (rn) be a sequence of real numbers and let B be the subset of R defined by B := { :ne N}. Let L E R and let f : B →R be defined by f (;) = an. Prove that lim a, = L if and only if lim f(x) = L. n00 Let c be a cluster point of a subset A of R and let f : A → R be a function. Suppose for every sequence (a,) in A such that lim a, = c, the sequence (f(am)) is a Cauchy sequence. Prove that lim f(x) exists. "/>
Extracted text: (a) Let A be the subset of R defined by A := {1-" : n e N}. Find all cluster points of A and justify your answer. (b) Let S C R and let c e R be a cluster point of S. Prove that for each e > 0 the set W := Sn (c- e, c+ e) has infinitely many elements. Let (rn) be a sequence of real numbers and let B be the subset of R defined by B := { :ne N}. Let L E R and let f : B →R be defined by f (;) = an. Prove that lim a, = L if and only if lim f(x) = L. n00 Let c be a cluster point of a subset A of R and let f : A → R be a function. Suppose for every sequence (a,) in A such that lim a, = c, the sequence (f(am)) is a Cauchy sequence. Prove that lim f(x) exists.