The cauchy-sequence tag has no wiki summary.
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Cauchy product of two geometric series
I have a question for my class that asks to find the product of two infinite series, namely $a_n = b_n = \sum\limits_{k=0}^\infty r^n$ for $r\in(0,1)$. That is find $c_n = \sum\limits_{k=0}^n a_k ...
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Proof of an intuition for equivalent definitions? (2)
I apologize for the double-post, but I asked the absolute wrong question and received a correct answer for that incorrect question. My apologies.
I have a very strong intuition about the ...
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Proof of an intuition for equivalent definitions?
I have a very strong intuition about the equivalency of the following definitions for Cauchy sequences:
$$\forall \varepsilon > 0, \exists M \in \mathbb{N} : k, l \geq M \implies d(x_k, x_l) < ...
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sequences of functions $\cos(\frac{x}{n})$
I have the sequence $(f_n)$ of functions $f_n: [-1, 1] \rightarrow \mathbb{R}$, defined by $f_n(x) = \cos\left(\frac{x}{n}\right)$. I need to show that $(f_n)$ is Cauchy in the space $(C[-1, 1], d)$ ...
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Determining Complete Metric Spaces
I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
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Correctness of Analysis argument with Cauchy sequences
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$
Proof $\rightarrow$
Suppose $(x_n) \to a$ and $(y_n) ...
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Cauchy Sequences and Analysis
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$.
What is the significance of $\mathbb{R}$ in this ...
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Need to prove $f$ continuous at $x_0$ iff for every monotonic sequence $(x_n)$ converging to $x_0$ we have $\lim f(x_n)=f(x_0)$
This was a problem that the Professor went over in class, but I am having trouble understanding and finishing the proof. The full question is:
$f:I \rightarrow \mathbb R$ is continuous at $x_0 \in I$ ...
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Is the set of integers Cauchy complete?
http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
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Cauchy sequence is convergent iff it has a convergent subsequence
Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence.
Note: ...
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Question in real analysis
I need help with this problem.
Show that a Cauchy sequence in $[0,1]$ must converge to a point of $[0,1]$.
Thank you
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55 views
Convergent & Cauchy Sequence related prove
(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $\{a_n\}\to a$ and $\{b_n\}\to b$.
Prove that $\{a_n+b_n\}\to a + b$
(2) Prove that a convergent sequence is Cauchy. ...
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Banach spaces and their unit sphere
Let $X$ be a normed vector space.
Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges.
Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
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Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?
Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.
Is $f$ continuous?
Let $f$ be ...
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How do I find the limit of the sequence $a_n=\frac{n\cos(n)}{n^2+1}$ and prove it is a Cauchy sequence?
I need to study the limit behavior of $a_n=\frac{n\cos(n)}{n^2+1}$, which can be written as $\frac{n}{n^2+1}\cos(n).$ I knew that it wasn't going to be monotone because $cos(n)$ oscillates between -1 ...
