Analysis 1 - Completeness Theory/Property/Axiom?

Question

Let {An} be an increasing sequence. Use the completeness property for increasing sequences to explain why {An} must tend to infinity if it does not converge to a finite limit.

Answer 1

What is the completeness axiom?

Simply this: if {An} is an increasing sequence in R, which is bounded above, and if {K} is the set of upper bounds, then a least upper bound exists for {K}.
This least upper bound will be lim An.

If there is no such K with the property,
then for every K > 0, there exists N(K) such that An > K
for all n > N.

This is the definition of the phrase "An -> infinity".

Answer 2

If the sequence is bounded above, then it would converge to a finite limit (equal to the least upper bound). Therefore, the sequence cannot be bounded above, so must tend to infinity.

Answer 3

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