How do you know if a sequence has a convergent subsequence?
You need to first show that the sequence is bounded so that way you know it has a convergent subsequence. by triangle inequality. and since n > n ? = max { n ? ?, n ? ? }, you know that n > n ? ?. Let ( a n) be a Cauchy sequence of reals. It is bounded [ There is an N such that a N, a N + 1, … are in ( a N ? 1, a N + 1).
What if the sequence Xn does not converge to X?
(a) Complete the following statement: “If the sequence xn, n = 1, 2, 3? does not converge to x as n ? ?, that means that there exists an ? > 0 such that...” (b) Consider the sequence xn = ( ? 1)n, n = 1, 2, 3? that is, the sequence is ( ? 1, 1, ? 1, 1, ? 1,...).
Is a sequence of real numbers convergent or bounded?
If the sequence of real numbers is convergent, then is bounded. A nondecreasing sequence which is bound above is convergent. A nondecreasing sequence which is not bound above diverges to infinity.
What is a sequence of nonnegative numbers that converges to zero?
Illustration: the sequence is a sequence of nonnegative numbers that converges to zero, and it doesn’t converge to any other limits. All subsequences of a convergent sequence of real numbers converge to the same limit. Note: if the subsequences of a sequence doesn’t converge to the same limit, then we can say that the sequence is divergent.