All Questions
Tagged with metric-spaces cauchy-sequences
426
questions
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64
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Prove that $(X, d)$ is a complete metric space where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ defined by...
I am given a metric space $(X, d)$ where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ the metric defined by
$$
d(x, y) = \begin{cases}(\sup\{n \in \mathbb{N}: x_k = y_k \...
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1
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34
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Completeness preserved under specific homeomorphism
Problem: Let $(X,d)$ be a complete metric space, $(Y,d')$ a metric space, and $f\colon X\to Y$ a homeomorphism, such that there exists a $c>0$, for which $$c\cdot d(x,y)\leq d'(f(x), f(y)).$$ Show ...
2
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1
answer
49
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Understanding a proof that a uniformly Cauchy sequence of continuous functions $\{f_n\}_n$ converges uniformly to a limit function $f$
Suppose $\{f_n\}_n$ is a uniformly Cauchy sequence of continuous function $f_n:\mathbb{R}\to S$ for a complete metric space $S$. I am trying to understand the "standard" proof that the ...
- 1,231
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2
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75
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Theorem 3.11 (c): Rudin's PMA
I wanted to ask some clarification on one of the proofs in Rudin's PMA, specifically Theorem 3.11 (c). It states that
In $\mathbb R^k$, every Cauchy sequence converges.
The proof for it is as ...
4
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1
answer
85
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Using only the universal property, prove that $X$ is dense in its Cauchy completion
Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
- 4,119
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1
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89
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Prove that if a cauchy sequence admits a convergent subsequence in a metric space, then the sequence converges to same limit
Problem: Let $(X,d)$ be a metric space and let $\{x_n\}_{n=1}^\infty$ a Cauchy sequence in $X$. Prove that if $\{x_n\}_{n=1}^\infty$ admits a convergent subsequence, then $\{x_n\}_{n=1}^\infty$ ...
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137
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Prove that $d(x_n,x_{n+1})→0$ $\iff$ $(x_n)$ ia a Cauchy sequence.
Let $X\neq\varnothing$ and let $d:X\times X→\mathbb{R}$ be a function such that
$$d(x,y)=0 \iff x=y,$$
$$d(x,y)=d(y,x),$$
$$d(x,z)≤\max\{d(x,y),d(z,y)\}.$$
Let us say that $d$ is a metric. Let $(...
- 73
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In a complete space $X$ is every $x \in X$ the limit of a sequence $\{x_n\}$ such that $x \not\in \{x_n\}$? [closed]
Let $X$ be a complete metric space. Then for any point $x \in X$, can it be shown that there exists a sequence $\{x_n\} \in X$ such that $x \not\in \{x_n\}$ and $x_n \rightarrow x$? More generally, I ...
- 6,295
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1
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45
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Confusing notation of sequences in $k$-dimensional Euclidean spaces.
Suppose $(x^{(n)})$ is a sequence in $\mathbb R^k$, $k \in \mathbb N$. From what I understand, $(x^{(n)})$ is a sequence of sequences $(x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, ..., x_k^{(n)})$. Or in other ...
- 309
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0
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57
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Is a Cauchy sequence in R necessarily a Cauchy sequence in Q
Let $(x_n)$ be a Cauchy sequence in the metric space $\mathbb{R}$ with the Euclidean metric, with the property that $x_n\in\mathbb{Q}$ for all $n\in\mathbb{N}$.
Is it true that $(x_n)$ in the metric ...
- 304
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1
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107
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A property of complete metric spaces makes them length (path or inner) metric spaces, Clarification of a proof
In the book "Metric Structures for Riemannian and Non-Riemannian Spaces", by Misha Gromov, I found a proof of the following statement (of Theorem 1.8. restated here more concentrated)
Let $(...
- 4,644
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49
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I need help understanting one of the implications of Cantor's Nested Set Theorem
Today in class we proved the following:
Let $(E,d)$ be a metric space.
E is Complete $\Rightarrow$ For every decreasing sequence $(A_n)_{n∈\Bbb N}$ of non empty subsets of E such that $\lim_{n\to\...
- 1
2
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Is the sequence Cauchy and convergent with the $d_2$ metric
Let the space $l^2$ be equipped with its usual metric , $d_2$. Let $x ∈ l^2$ and consider the following sequence $(z_n)^∞_{n=1}$, where
$z_1= x = (x_1,x_2,x_3,...)$
$z_2 = (x_1/2,x_2/2,x_3/2,...)$
$...
- 23
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1
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60
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proof verification for completeness of a metric space $(M,f)$ where $M=X \times Y$ and $f((x_1,y_1),(x_2,y_2))=d(x_1,x_2)+e(y_1,y_2)$
The question I've been given is as follows: Given that $(X,d),(Y,e)$ are complete metric spaces, show that the metric space $(M,f)$ is complete where $M=X \times Y$ and $f:M \times M\rightarrow \...
user1167555
2
votes
2
answers
307
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Let $(X,d)$ be a complete metric space.Prove $(X,d')$ is a complete metric space $d'(x,y)=\begin{cases}\max\{1,d(x, y)\}&x\neq y\\0&x=y\end{cases}$ [duplicate]
Let $(X,d)$ be a complete metric space.
Prove $(X,d')$ is a complete metric space such that $d'(x,y)=\begin{cases} \max\{1, d(x, y) \}&x\neq y\\0&x=y\end{cases}$
Can't figure out why this ...
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