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Prove that if

In your own words, what does the Bolzano-Weierstrass Theorem say and why is it important/where is it employed?

Prove that the series

What is a quick way to tell if a series diverges?

Prove that the sequence defined by and 

A sequence that is bounded but not convergent.

Two series

An uncountable set that is a proper subset of

In your own words, what does it mean if two sets have the same cardinality?

Prove that every convergent sequence is bounded.