What is the relationship between "existence limit", "sequence convergence" and "boundedness"? Such as the title What is the relationship between the so-called "boundary" and "limit"? Must the boundary be greater than the limit?

What is the relationship between "existence limit", "sequence convergence" and "boundedness"? Such as the title What is the relationship between the so-called "boundary" and "limit"? Must the boundary be greater than the limit?

Of course, there is a limit in the convergence of a sequence, which is equivalent to the two statements. If a sequence converges, then it must be bounded, and vice versa!
For example: xn = 1, - 1,1, - 1
|Xn|

How to prove that a bounded nonconvergent sequence must have two subsequences which converge to different limits?

It is proved that: if any convergent subsequence (which must exist) is set as a, then there must be an infinite term (which is still bounded) of the bounded sequence outside a sufficient small field of a, so that the limit of convergent subsequence is not equal to a

The convergence proof of high number and sequence
If the odd and even subsequences of a sequence {xn} converge to a, then please prove that {xn} also converges to a

Use the definition
For any ε > 0, there is a corresponding K1, such that for any k > K1, a (2k) - a │ K2, a (2k + 1) - a │