We need first a definition
Definition Letbe a set of real numbers. A real numberis an accumulation point ofif and only if every neighbourhood of contains infinitely many points of
For example 0 is an accumulation point of the setsincecontains allforNote the accumulation point need not belong to the set.
The Bolzano Weierstrass Theorem
Every bounded infinite set of numbers has at least one accumulation point.
Proof: Letbe a bounded infinite set. Sinceis bounded, there are real numbersand %beta such thatIfis the midpoint of this interval, then at least one of the sets must contain an infinite number of members ofChoose one with this property and call itIfis the midpoint of this interval, then at least one of the setsmust contain an infinite set of members ofChoose one with this property and call itContinuing in this fashion we obtain for each positive integera closed intervalwith the following properties:

contains infinitely many points of
Sincefor allthe setis bounded. Letis the desired accumulation point. We need to show that every neighbourhood ofcontains infinitely many points ofLetbe any neighbourhood ofthen there existssuch thatNowis not an upper bound forsinceis the least upper bound hence there is a positive numbersuch thatIfthen by the construction aboveEach intervalcontains infinitely many points ofso the proof would be complete ifcould be found such thatAs noted above,for so it will be enough to chooselarge enough so thatandthenThuscontainshencecontains infinitely many members ofandis an accumulation point of
The boundedness of the set is necessary because for exampleis infinite but has no accumulation points.