Every bounded sequence is NOT necessarily convergent.

Does a bounded series always converge?

No, a bounded series does not necessarily converge.

Do all bounded monotone sequences converge?

Not all bounded sequences, like (−1)n, converge, but if we knew the bounded sequence was monotone, then this would change. if an ≥ an+1 for all n ∈ N. A sequence is monotone if it is either increasing or decreasing. and bounded, then it converges.