Prove from definition that the sequence a_n = 5 - 1/n^2 converges.
Prove from definition that the sequence a_n = 1/n^2 is Cauchy.
Prove every Cauchy sequence is bounded (without looking at the book's proof). You may not assume that every Cauchy sequence converges.
Prove every convergent sequence is Cauchy (without looking at the book's proof).
The proof of Corollary 2.13 given in the book is missing a lot of details (in fact, most of the proofs in the book leave out too much). Give a complete proof by adding all the missing details.
Prove every decreasing sequence that's bounded below converges (without looking at the book's proof).