Theorem:
Closed Sets, Accumulation Points, and Sequences
A set
S
R
is closed if and only if every Cauchy sequence of elements in
S
has a limit that is contained in
S
.
Every bounded, infinite subset of
R
has an accumulation point.
If
S
is closed and bounded, and
is any sequence in
S
, then there exists a subsequence
of
that converges to an element of
S
.
Proof:
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