If f E rightarrow R is a continuous function defined on some

If f: E rightarrow R is a continuous function defined on some set E R and if {a_j}_j N E is a Cauchy sequence is it true that {f(a_j)}_j epsilon N is also a Cauchy sequence ? If true prove it. Otherwise provide a counterexample.

Solution

Solution: Not Necessarily,

{a_j} =1/n this converges with limit 0, and cauchy sequence , but if we take function which reciprocal sequence

f(x)=1/a_j  then sequence is n which is not convergent. hence not cauchy