Can someone explain to me how I can go about solving part (a) and part (b) (e.g.
ID: 2979400 • Letter: C
Question
Can someone explain to me how I can go about solving part (a) and part (b) (e.g. theorems I should be using...concepts I should know)? I'm looking for a detailed explanation.
Explanation / Answer
A sequence x_n is NOT a Cauchy sequence provided there is some ?> 0 such that, for every natural number N, there are integers m, n > N with |x_n - x_m| ? ?. It seems to me that this is very simple. A Cauchy sequence is a sequence such that |x_m - x_n| ? 0 as the indices m,n ? oo. In other words, the terms get arbitrarily close the larger the indices are. A sequence which is not a Cauchy sequence lacks this property. This means that if a sequence is not a Cauchy sequence the terms do not get arbitrarily close no matter how large the indices are. I have rephrased this to use ? as a measure of closeness; this must fail if ? is sufficiently small no matter how large m & n are. That is, given any (large) N, there are even larger m & n with the distance between x_m & x_n at least as big as ??. (|x_m - x_n| ? ?