Let C Be A Cluster Point Of A R, And F: A Rightarrow R Be A Function. Suppose For Every Sequence {Xn} In A, Such That Lim Xn=C, The Sequence {F(Sn)} Infinity N=1 Is Cauchy. Prove That Limx Rightarrow C F(X) Existe

 

 

 

 

and g: BR are functions such that f(x) c2 as x c and g(y) and show h(x) Las xC1. 2. h(x) == 8 (f(x)) Exercise 3.1.10: Let c be a cluster point of ACR, and f: AR be a function. Suppose for every sequence {xn} in A, such that lim x = c, the sequence {f(xn)}1 is Cauchy. Prove that limxc f(x) exists. Exercise 3.1.11: Prove the following stronger version of one direction of Lemma 3.1.7: Let SCR, c be a cluster point of S, and f: SR be a function. Suppose that for every sequence {xn} in S\{c} such that