Convergence sequence of Higher Mathematics Known a How are those two formulas simplified?

Convergence sequence of Higher Mathematics Known a How are those two formulas simplified?

The title is wrong. It should be | xn-a|

What is the relationship between boundedness and limit in Higher Mathematics? Is it possible to converge if there is a limit?

Boundedness means that there are maxima, minima or infinitely close to them
For example, 1,0,1,0,1,0 This is called the sequence bounded
To have a limit is to go to infinity, and infinity is close to a value
For example, 1,1 / 2,1 / 3,1 / 4 The sequence tends to 0 and the limit is 0
The third question is not very clear

Bounded does not necessarily converge, convergence must be bounded, why

The odd term is equal to - 1, and the even term is equal to 1. This sequence is bounded, but it does not converge. The following is the proof that convergence must be bounded
The purpose is to prove the boundedness of convergent sequence. Sequence {xn} converges to A. according to the definition of limit, for any E > 0, there exists positive integer n. when n > N, the inequality / xn-a / < e holds, where e can be selected as 1. Intuitively, when n tends to infinity, the value of XN is infinitely close to A. in order to describe this property accurately, n is introduced, That is, xn is infinitely close to A. after n > N, all xn are less than a plus a positive number (E). It is proved that the sequence of numbers starting from n is bounded (all less than e + | a |)