- #1
kingwinner
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Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I'm really not having much progress on this question. My thoughts are as shown above.
boboYO said:2) since xn does not converge to a, then for some epsilon, say e, there are infinitely many points in the sequence xn such that |x-a|>e. So, make the first element of our subsequence the first such point, the 2nd element the 2nd such point, and so on. We won't run out of points because there are infinitely many of them.
3) just making sure I'm clear here, i mean that all subsequences of w can not converge to a. think about it, draw a diagram if you have to: if you have a sequence that is always at least a certain distance away from a then obviously no subsequence of it can converge to a. I edited my original post, hopefully a bit clearer now.
it's quite straightforward. just find a suitable e.xn does NOT converge to a iff
there exists e>0 s.t. for all N, there exists n s.t. n>N, but |xn -a|>=e.
In a metric space, a sequence {xn} is said to converge to a point x if for every positive real number ε, there exists a positive integer N such that for all n > N, the distance between xn and x is less than ε.
A subsequence of a sequence is formed by selecting certain terms from the original sequence in a specific order. The convergence of a subsequence means that the selected terms of the subsequence converge to a point, while the convergence of a sequence means that all terms of the sequence converge to a point.
Yes, a subsequence can converge to a different point than the original sequence. This is because the subsequence may consist of a different selection of terms, which can lead to a different point of convergence.
If a sequence converges, then all of its subsequences also converge to the same point. However, the converse is not true. A subsequence may converge even if the original sequence does not converge.
Yes, a subsequence of a divergent sequence can still converge to a point. This is because the subsequence may consist of a different selection of terms that may exhibit a different behavior and lead to convergence.