Ch. 6 More on limits of sequences

Question 1

Definition of monotone increasing

Answer 1

A sequence is monotone increasing if xn ≤ x(n+1) for all n in the natural numbers

Question 2

Proposition 6.4 (subsequences)

Answer 2

If xn is convergent with limit x, then a subsequence, xnj is also convergent with limit x

Question 3

State the Bolzano-Weierstrass theorem

Answer 3

Let xn be a bounded real sequence, then xn has a subsequence that is convergent

Question 4

Theorem 6.8 and 6.9

Answer 4

If xn is a Cauchy sequence, it is also bounded

If xn is a convergent sequence, then it is also a Cauchy sequence