Let $(X,d)$ be a metric space. If $K⊆X$, and $K$ is a closed set. Does that mean any Cauchy sequence in $K$ converges in $K$?
If no, could someone give an example?
If yes, then what is the difference between complete sets and closed sets?
2026-04-02 07:31:23.1775115083
Closed Sets VS. Complete Sets
1.5k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in CAUCHY-SEQUENCES
- Closure and Subsets of Normed Vector Spaces
- Proof check - If two sequences A and B are equivalent, then the first one is a Cauchy sequence if and only if the second one is a Cauchy sequence too
- Proof check - The multiplication of two real numbers is a real number (Cauchy sequences)
- If $\|x_k - x_{k-1}\| \le \frac{1} {k^2}$, what is the limit of $\lim_{k \to \infty} (k x_k - (k-1) x_{k-1})$?
- Prove that $f$ has a fixed-point $x_0$ with $x_0 > 0$
- Proving that the sequence $\{\frac{3n+5}{2n+6}\}$ is Cauchy.
- Why can't all pointwise continuous functions preserve Cauchy sequences?
- Proving that sequence with given criteria is Cauchy?
- Determining whether sequence $a_n=\sum_{k=1}^n\frac n{n^2+k}$ is convergent or not.
- How does the Cauchy criterion for double series imply that the comparison test can be applied to double series if the terms are nonnegative?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The rationals are closed in the rationals, but obviously not complete. For strict inclusion, take $\mathbb{Q} \cap [0,1]$ in $\mathbb{Q}$. Also not complete, but closed.
Incidentally, complete implies closed. Just look at limit points and sequences.