It is proved that if a sequence converges to a, then any of its subsequences converges and the limit is a Can that be expressed in mathematical language?

It can be proved to the contrary
If the subsequence does not converge or the convergence limit is different from a, then the original sequence does not converge

RELATED INFORMATIONS

1. Prove the convergence of sequence √ 2, √ (2 + √ 2), √ (2 + √ (2 + √ 2)),. And find its limit
2. How to find the sum of the first n terms of an = 1 / n
3. One X of E, X tends to 0 - why is the limit 0?
4. Finding limit: when x tends to infinity, find the limit of (x + 1) &# e ^ X Limit: when x tends to infinity, find the limit of (x + 1) / e ^ X
5. (e ^ n times n!) / N ^ n how to find the limit x tending to infinity
6. X tends to 0, the limit of sin1 / x ^ 2-1 / sin2x (not SiNx ^ 2, sin2x)
7. Functions are bounded, unbounded, convergent, divergent, with and without limits. What are the relations between these relations?
8. Find the left and right limit of function f (x) = [x] at x = 0
9. What function is f (x) = sgnx? What is the image like?
10. What is the intermediate value principle of continuous function?
11. On N > N in sequence limit For example, if the sequence is n + 1 / N and the limit is 1, 1 / N can be calculated If you want the distance between the sequence and the limit to be less than 0.001, as long as n > 2000 In the concept of limit, there is an n > N, that is, 2001 > 2000 What does this n stand for?
12. Why convergence sequence must be bounded sequence? Don't go too far But I hope it makes me understand
13. Is the sequence with limit necessarily convergent? Is it not necessarily bounded
14. A question about the proof that the convergence of sequence of higher numbers must be bounded This theorem is proved in page 40 of the fourth edition of Tongji, because the sequence {xn} converges, let limxn = a, according to the definition of sequence limit, there exists positive integer n for ε = 1, such that for all xn when n > N, when the inequality | xn-a | n, | xn | = | (xn-a) + a | ≤ | xn-a | + | a|
15. Convergence sequence of Higher Mathematics Known a How are those two formulas simplified?
16. When we prove the limit of convergent sequence in higher numbers, let ε 0 (...), as long as 1 / (4N + 2) 1 / 4 (1 / ε - 2), the inequality | xn-a | n has | (3N + 1) / (2n + 1) - 3 / 2|
17. 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?
18. There is this formula in Higher Mathematics: prove that the complement set of a intersection B = a complement and B complement How to prove this formula
19. What's the difference between a and B
20. How can higher mathematics be divided into a, B, C, D, e, f? What do they stand for