I am studying Classical Analysis with Marsden book. At very first chapter it covers sequence, field, etc... The book has theorems 1."Let F be an ordered field. We say that the monotone sequence property if every monotone increasing sequence bounded above converges." 2."An ordered field is said to be complete if it obeys the monotone sequence property" 3."There is a unique complete ordered field called the real number system." so I wonder for 1. as I know monotone increasing sequence has a definition of A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ 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.