Definition of cauchy sequence9/13/2023 ![]() Proof.Sincefangforms a Cauchy sequence, for 1 there existsN2Nsuch that jan amj<1 8n m N: In particular, jan aNj<1 8n N: Hence ifn N, then janj jan aNj+jaNj<1 +jaNj 8n N: LetM maxfja1j ja2j ::: jaN1j 1 +jaNjg. ![]() ![]() Which is precisely the condition for a sequence to be Cauchy. Lemma 1.0.3.Iffangis a Cauchy sequence, thenfangis bounded. A sequence (xn) of real numbers is a Cauchy sequence if for each k in Z+ there exists Mk in Z+. $$ d(x_n, x_m ) 0$, let $N$ be large enough that $\frac$, if $n,m\geq M$ then $d(x_n,x_m)< \epsilon$, I'm going through the proof of Banach Fixed Point for a metric space $(X,d)$ in this pdf, the step I having trouble with in is the one after the geometric series is summed: The relation in Definition 10.1 is an equivalence relation on the set of all rational Cauchy sequences.
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