Completion Notes

Next if xn nin v we dene the equivalence class of xn

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Unformatted text preview: V =0 This definition is rigged so that any two convergent sequences have the same limit if and only if they are equivalent. Next, if {xn }n∈IN ∈ V , we define the “equivalence class of {xn }n∈IN ” to be the set {xn }n∈IN = {yn }n∈IN ∈ V {yn }n∈IN ∼ {xn }n∈IN of all Cauchy sequences that are equivalent to {xn }n∈IN . We shall shortly prove September 7, 2011 Completion 1 Lemma 3 ∼ is an equivalence relation(1) . In particular, if {xn }n∈IN , {yn }n∈IN ∈ V then either {xn }n∈IN = {yn }n∈IN {xn }n∈IN ∩ {yn }n∈IN = ∅ or If you think of a Cauchy sequence as one person and an equivalence class of Cauchy sequences as a “family” of related people, then the above Lemma says, that the whole world is divided into a collection of nonoverlapping families. Next, we define H= {xn }n∈IN ∈ V {xn }n∈IN as the set of all “families” and prove Lemma 4 If {xn }n∈IN , {yn }n∈IN ∈ V then lim xn , yn n→∞ V exists. Lemma 5 Define, for each {xn }n∈IN ∈ H, {yn }n∈IN ∈ H and α ∈ C, {xn }n∈IN + {yn }n∈IN = {xn + yn }n∈IN α {xn }n∈IN = {αxn }n∈IN {xn }n∈IN , {yn }n∈IN H = lim xn , yn n→∞ V Each of these operation...
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