True or False If true prove it carefully If false provide a

True or False? If true, prove it carefully. If false, provide a counterexample and explain it. If (a_n) is any Cauchy sequence of real numbers and (b_n) si any increasing sequence of negative numbers, then the sequence (a_n - 2b_n) converges.

Solution

False.

. If (xn) is a sequence then limn xn = , or xn as n , if for every M R there exists N R such that xn > M for all n > N. Also limn xn = , or xn as n , if for every M R there exists N R such that xn < M for all n > N. That is, xn as n means the terms of the sequence (xn) get arbitrarily large and positive for all sufficiently large n, while xn as n means that the terms get arbitrarily large and negative for all sufficiently large n. The notation xn ± does not mean that the sequence converges.