The book from which I am learning analysis states cantor's completeness principle as follow. ;
"Consider a nest of closed intervals I1,I2,I3...In , each being denoted as [an,bn]. As n tends to infinity .. the common intersection of all the intervals is a point which is equal to the limit to which both an and bn converges to."
I can prove the converegence of the sequences an and bn using Weierstrass principle.
Now for the two sequences we can see that for any two given positive integers m and n a(m)>b(n)...........(1)
Now what I want to know is that does the cantor's principle imply that given any two sequences ,one of which is monotonically increasing and another is monotonically decreasing then the two sequences converge to the same limit ? Obviously this is wrong, and there is a gaping hole in my understanding. Can I know where my understanding misses the right track ? Secondly, how to put this principle into use, when we just know that there exist a common limit for both the sequences but don;t know its value?