Intereting Posts

Bartoszyński's results on measure and category and their importance
Quadratic Polynomial factorization
How to explain the formula for the sum of a geometric series without calculus?
Is every submodule of a projective module projective?
Compute probability of a particular ordering of normal random variables
Explain why perpendicular lines have negative reciprocal slopes
Prove that $\sum_{i=0}^{63}f_{i}\cdot\left(n+i\right)^{5}=0$
Does every prime divide some Fibonacci number?
Limit points and interior points
Superelliptic Area Of $x^5+y^5=r^5$
Inequality involving sums of fractions of products of binomial coefficients
An exercise on liminf and limsup
Number of straight line segments determined by $n$ points in the plane is $\frac{n^2 – n}{2}$
Square roots of $-1$ in quaternion ring
Converge or diverge? : $\sum_{n=1}^{\infty}\frac{\tan{n}}{2^{n}}$

Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N >0$ such that for all $n,m > N$, we have

$$d(x_n,x_m) < \frac{\epsilon}{2}.$$

By hypothesis, we can take also $K >0$ such that for all $n_k > K$, we have

- Proof that the solution to cosx = x, is the limit of a recursive sequence.
- I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.
- Confused with proof that all Cauchy sequences of real numbers converge.
- Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
- The space of continuous functions $C()$ is not complete in the $L^2$ norm
- If $\lim a_n = L$, then $\lim s_n = L$

$$ d(x_{n_k},x) < \frac{\epsilon}{2}.$$

Put $M = \max \{N,K\}$. Therefore, for all $n,m,n_k > M$, we have

$$ d(x_n,x) \leq d(x_n, x_{n_k}) + d( x_{n_k},x) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$

Hence, $x_n \to x$ as desired.

Is this a correct approach? Thank you very much in advance.

- Weakest topology with respect to which ALL linear functionals are continuous
- Need help with understanding the Basis for a Topology
- Closed image of locally compact space
- Domain is Hausdorff if image of covering map is Hausdorff
- How to get intuition in topology concerning the definitions?
- Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?
- Every order topology is regular (proof check)
- How to prove that a set R\Z is open
- Indiscrete rational extension for $\mathbb R$ (examp 66 in “Counterexamples in topology”)
- Understanding the definition of a compact set

Very nice.

As an alternative, you could say that $d(x_{n_k},x)<\epsilon/2$ whenever $k>K.$ Then, you want to put $M=\max\{N,n_K\}.$ You know that $k>K$ if and only if $n_k>n_K,$ which allows you to draw the desired conclusion.

- Finding the limit of $(1-\cos(x))/x$ as $x\to 0$ with squeeze theorem
- Obstruction to the splitness of an exact sequence of holomorphic vector bundles
- What Is Exponentiation?
- Prove: $(a + b)^{n} \geq a^{n} + b^{n}$
- How to find the value of $\sin{\dfrac{\pi}{14}}+6\sin^2{\dfrac{\pi}{14}}-8\sin^4{\dfrac{\pi}{14}}$
- Maybe is right $\frac{n^2 + 1}{4k + 3} \notin \mathbb{Z}, n, k \in \mathbb{N}^{+}$
- If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{*}$ is a root of $x^2-1$. Prove that $n$ divides 24.
- Applying Central Limit Theorem to show that $E\left(\frac{|S_n|}{\sqrt{n}}\right) \to \sqrt{\frac{2}{\pi}}\sigma$
- Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.
- $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$
- “Proof” that $g(t) = (t,t,t,…)$ is not continuous with uniform topology
- how to prove $\sum_{k=0}^{m}\binom{n+k}{n}=\binom{n+m+1}{n+1}$ without induction?
- Transforming linear combination of the cosine and sine function
- Proving that a doubly-periodic entire function $f$ is constant.
- Positive operator is bounded