Let (an) be a bounded sequence, and define the set S = {x ∈ R : x < an for infinitely many terms an}. Show that there exists a subsequence (ank) converging to s = supS. (This is adirect proof of the Bolzano–Weierstrass Theorem using the Axiom ofCompleteness.)
Let (an) be a bounded sequence, and define the set S = {x ∈ R : x < an for infinitely many terms an}. Show that there exists a subsequence (ank) converging to s = supS. (This is adirect proof of the Bolzano–Weierstrass Theorem using the Axiom ofCompleteness.)
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Let (an) be a bounded sequence, and define the set
S = {x ∈ R : x < an for infinitely many terms an}.
Show that there exists a subsequence (ank) converging to s = supS. (This is a
direct proof of the Bolzano–Weierstrass Theorem using the Axiom of
Completeness.)
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