I am studying Classical Analysis with Marsden book.(adsbygoogle = window.adsbygoogle || []).push({});

At very first chapter it covers sequence, field, etc...

The book has theorems

1."Let F be an ordered field. We say that themonotone sequence propertyif every monotone increasing sequence bounded above converges."

2."An ordered field is said to becompleteif it obeys the monotone sequence property"

3."There is a unique complete ordered field called thereal number system."

so I wonder

for 1. as I know monotone increasing sequence has a definition of

A sequence (an) ismonotonic increasingifan+1≥anfor alln∈N

then what about e.g (an) = n , a1=1 a2=2 a3=3 .... ? this sequence is monotonic increasing but not bounded.

contradicts to 1.

In the same reason, according to 2 and 3. Any sequences in real number system has to obey monotone sequence property and it means it has to be bounded. But I can come up with lots of sequence which has real numbers as it's element that explodes.

like (an) = 2n , or Harmonic sequence.

I am pretty sure I have a critical misconception. but don't know what it is. help me please.

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# I Completeness Property (and Monotone Sequence)

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