Sequences (monotonic, non increasing, or non decreasing)

Question

What does it mean/ what are cases where M (value at which a sequence is bounded*) is not equal to the limit? & can a sequence be bounded and not have a limit?

 

*separate answers for bounded from above and below please.

 

Is there a way to find whether a function nondecreasing, nonincreasing or monotonic without graphing it or trying to plug in numbers (because if you don't try the right numbers you might miss a decrease or increase)?

 

Are negative values for x not looked at when dealing with sequences?

 

Thank you! 

Answer 1

An oscillating sequence could have both an upper bound and a lower bound (ALL values would lie between these two different bounds) but it need not converge to a fixed Real limit.