Prove that every sequence in a discrete metric space converges
and is a Cauchy sequence.
This is all that was given to me... so I am unsure how I am
supposed to prove it....
Recall that a sequence an is Cauchy if, given ε > 0, there is
an N such that whenever m, n > N, |am − an| < ε.
Prove that every Cauchy sequence of real numbers converges.
Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >
1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the
sequence is an integer plus or minus 1/3 ). Show that the sequence
sn is eventually constant, i.e. after a point all terms of the
sequence are the same
Use the Cauchy Criterion to prove the Bolzano–Weierstrass
Theorem, and find the point in the argument where the Archimedean
Property is implictly required. This establishes the final link in
the equivalence of the five characterizations of completeness
discussed at the end of Section 2.6.