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operationsres

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http://www.u.arizona.edu/~mwalker/econ519/Econ519LectureNotes/Bolzano-Weierstrass.pdf

I'll type out the proof and cease typing it at the part that I don't understand.

**The Bolzano-Weierstrass Theorem**: Every bounded sequence of real numbers has a

convergent subsequence.

**Proof**: Let [itex]\{x_n\}[/itex] be a bounded sequence and without loss of generality assume that every term of the sequence lies in the interval [itex][0,1][/itex]. Divide [itex][0,1][/itex] into two intervals, [itex][0,0.5][/itex] and [itex][0.5,1][/itex]. (Note: this is not a partition of [itex][0,1][/itex].) At least one of the halves

contains infinitely many terms of [itex]\{x_n\}[/itex], denote that interval by [itex]I_1[/itex], which has length 0.5,...

**My misunderstanding**

The part that I don't understand is "At least one of the halves contains infinitely many terms of [itex]\{x_n\}[/itex]" ... Why can't [itex]\{x_n\}[/itex] be [itex]\{0.1,0.2\}[/itex], in which case one of the intervals contains 2 terms and the other contains 0 terms (with neither containing infinitely many)? No where did they state that [itex]\{x_n\}[/itex] couldn't have finitely many terms..?