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I understand (to the best of my ability) the Least Upper Bound property and the Nested Interval property, but I don't see how the two are equivalent properties.

LUB:

If [tex]S \subset \mathbb{R} [/tex] has an upper bound, then [tex]S[/tex] has a LUB

Nested Intervals:

If [tex]I_1 \supset I_2 \supset \cdots \supset I_n \supset \cdots[/tex] is a sequence of nested, closed, bounded, non-empty intervals, then

[tex]\bigcap_{n=1}^{\infty}I_n \neq \emptyset[/tex]

and length([tex]I_n \rightarrow 0[/tex], then [tex]\exists[/tex]

[tex]x_o \in \bigcap_{n=1}^{\infty}I_n[/tex]

(ie: a unique point exists in all the intervals)

Thanks in advance for clarifying this for me.

dogma

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# LUB and Nested Interval Equivalancy

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