Subsequence Definition and 61 Threads

Homework Statement Consider the sequence {x_k} = {(arctan(k^2+1),sink)} in R^2. Is there a convergent subsequence? Justify your answer. Homework Equations Every bounded sequence in R^n has a convergent subsequence. The Attempt at a Solution To show {x_k} is bounded: The range...

Homework Statement Consider the sequence \left\{ x_{n} \right\}. Then x_{n} is convergent and \lim x_{n}=a if and only if, for every non-trivial convergent subsequence, x_{n_{i}}, of x_{n}, \lim x_{n_{i}}=a. Homework Equations The definition of the limit of a series: \lim {x_{n}} = a...

Homework Statement Suppose that {Xn} is a sequence in R. Prove that Xn converges to a if and only if every subsequence of Xn converges to a. Homework Equations The Attempt at a Solution Let e>0, choose N in N st n >=N implies |Xn-a| <e. Since a subsequence, nk, is in N and...

According to the Bolzano-Weierstrass theorem, a bounded sequence has a convergent subsequence. My problem is with the proof. Either I've got a bad textbook, or my reading comprehension is lacking. This is how it's formulated: Let x1, x2, ... be a bounded sequence. Let E be the set of all...

Is the sequence defined as the denumeration of Q the only such sequence?

How can you deduce that nad bounded sequence in R has a convergent subsequence?

Homework Statement Prove: Let (Xn) be a sequence in R (reals). Then (Xn) has a monotone subsequence. Homework Equations Def: Monotone: A sequence is monotone if it increases or decreases. The Attempt at a Solution I know it has something to do with peak points...that is there...

Homework Statement Let {x_n} be a sequence in a metric space such that the distance between x_i and x_{i+1} is epsilon for some fixed epsilon > 0 and for all i. Can it be shown that this sequence has no convergent subsequence? Homework Equations None. The Attempt at a Solution...

Hello, here is the exercise from the book: ---------- Let (a_{n})_{n=0}^{\infty} be a sequence which is not bounded. Show that there exists a subsequence (b_{n})_{n=0}^{\infty} of (a_{n})_{n=0}^{\infty} such that the limit \lim_{n\rightarrow \infty} 1/b_{n} = 0. (Hint: for each natural...

Hi, Here is the question: Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity. To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner. My thoughts are: every convergent sequence has a convergent...

Does a subsequence only have to have "some" terms This is an example from my text, which I do not understand. Suppose (s_n)is a sequence of POSITIVE numbers such that inf{s_n | n in NaturalNumbers} = 0. The sequence need not converge or even be bounded , but it has a subsequence that...