Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy sequence in $ X $ is a Cauchy sequence in $ […]

Hi everyone I’m having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I’m wondering if the next reasoning is correct or maybe needs some changes: Definitions: Two sequence are equivalence $\iff$ $(\forall \varepsilon \in \mathbb{Q}^+\,) ( \, \exists N\in \mathbb{N}\,) \text{ s.t. […]

Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{2n-1})$ and $(a_{2n})$ has no Cauchy subsequence. Is it also true that $(a_n)$ has no Cauchy subsequence? Let $A=\{a_{2n}:n\in\mathbb{N}\}$, is it true every Cauchy sequence in $A$ is constant?

Given a compact metric space $(X,d)$, we consider $Iso(X,d)$ with metric $\rho$ such that $\lim _{n \rightarrow \infty} \rho(h_n, h) =0 \iff \forall x \in X: \lim _{n \rightarrow \infty} d(h_n(x), h(x))=0$. Could you tell me how to prove that if we have a $\rho$-Cauchy sequence of functions $h_n \in Y$, then for each $x\in […]

Suppose $(X,\tau)$ is a topological space and that $(X^2,\tau_2)$ is the product space. Now define $\mathscr S\!_\tau=\{W\in\tau_2|\Delta X^2\subseteq W\}$, where $\Delta X^2=\{(x,y)\in X^2|x=y\}$, and suppose $(x_n)$ is a sequence in $X$ such that $\forall S\in\!\mathscr S\!_\tau\exists N\in\mathbb N:(n,m>N\implies(x_n,x_m)\in S)$. Is it then possible to deduce that $(x_n)$ has a clusterpoint in $X$, $\exists x\in \!X\;\forall […]

i want to show that $\mathbb{R}$ with the following metric : $d_1(x,y)=|x^3-y^3|$ is complete. I think a good way to show it is to show that a sequence which is Cauchy for $d_1$ will also be Cauchy for the usual metric $d(x,y)=|x-y|$ but i’m not able to write it properly. More precisely i want to […]

So I’ve got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $\cos x = x$. There is also a small hint saying I should prove its a Cauchy sequence and then use MVT. Well, I’ve proven its Cauchy, […]

The title is kind of misleading because the task actually to show Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy without knowing that: Every bounded non-empty set of real numbers has a least upper bound. (Supremum/Completeness Axiom) A sequence converges if and only if it is Cauchy. (Cauchy Criterion) Every monotonic increasing/decreasing, bounded and […]

I have this problem: give an example of a real sequence $\;\{a_n\}\;$ with $$\lim_{n\to\infty}\left(a_{n+1}-a_n\right)=0\;,\;\;\text{but}\;\;\lim_{n\to\infty}a_n\;\;\text{doesn’t exist finitely}$$ The two classical examples I know, namely the harmonic series’s partial sums sequence, and the sequence $\;\{\log n\}\;$ , both use either infinite series theory or the continuity of the function $\;\log x\;$ , needed to deduce the existence […]

In many of the standard textbooks discussing Real Numbers, the Cauchy sequence that converges to $\sqrt{2}$ is given as 1, 1.4, 1.41, 1.414, 1.4142, … or 2, 1.5, 1.42, 1.415, 1.4143, … My question is how are these sequences generated? In other words, if I have 1, 1.4, 1.41 how do I figure out that […]

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