Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

1,514 questions
77 views

Does Cauchy Completeness imply the Heine-Borel theorem generally?

I've been working through some reverse math with the completeness definitions of a metric space. More over, I've learned that in a metric space X that is ordered, The Least Upper Bound Property, ...
59 views

If $\sum\limits^{\infty}_{n=1}|x_n|^2<\infty$ then $\{x_n\}$ is a Cauchy sequence

Let $\{x_n\}$ be a sequence of real numbers. How do I prove that if $\sum_{n=1}^\infty|x_n|^2<\infty$ then $\{x_n\}$ is a Cauchy sequence ? Is the converse true? How do I prove this? For the ...
39 views

Integral Approximation to Infinite sum Vs Cauchy's first theorem

$$\lim_{n\to \infty}\sum_{i=0}^n 1/(3*n +i)$$ . After applying cauchy's first theorem on pints, I get the answer as 1/4 , but after expressing the above sum as a definite integral I get the answer ...
20 views

47 views

61 views

Prove this sequence converges or diverges to $-\infty$

Let $a_n$ be a sequence such that for every $n$: $a_n\le\frac{1}{2}(a_{n-1}+a_{n-2})$. Prove that $a_n$ either converges to a real number $L$ or diverges to $-\infty$ $(L\in[-\infty,\infty))$. ...
96 views

Comparison Test for complex series: logical argument?

I am trying to show that the comparison test holds for complex series, meaning: if $\sum_{n=0}^{\infty} z_n$ is a complex series and $\sum_{n=0}^{\infty} a_n$ is a convergent non-negative real ...
51 views

Convergence of a sequence defined by a recurrence inequality [duplicate]

Let $(u_n)$ be a sequence of real numbers with the following property \begin{eqnarray*} \forall\,m,n\ge1,\quad u_{m+n}\le \frac{m}{m+n}u_m+\frac{n}{m+n}u_n \end{eqnarray*} Is it true that $(u_n)$ ...
39 views

Meaning of being a Cauchy sequence

A Cauchy sequence in a metric space $(X,d)$ is a sequence for which the distance between two terms can be made as small as we want, provided we look far enough in the sequence. Let $X \subseteq Y$, ...
55 views

54 views

Sequence of function on $\mathbb{R}$ Cauchy iff convergent

Theorem: Let $(f_n)$ be a sequence of functions on $I \subseteq \mathbb{R}$. Then $(f_n)$ pointwise convergent iff pointwise cauchy. Here, I only prove "$\Longleftarrow$" since the converse is very ...
66 views

Proof verification of $x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$ diverges by the negation of Cauchy

Let $\{x_n\}$ denote a sequence: $$x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$$ Show that $\{x_n\}$ diverges using the negation of Cauchy criterion. I would like to kindly request a ...
59 views

Prove that $\{x_n\}$ is a Cauchy sequence iff $\forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$

Let $\{x_n\}$ denote a sequence. Prove that: $$\{x_n\}\ \text{is fundamental} \iff \forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$$ Let $P$ be a ...
85 views

Is the notion of Cauchy sequences definable in a bornological topological space?

Being a Cauchy sequence is not a topological property, i.e. two metrics can induce the same topology and yet a sequence which is Cauchy in one may not be Cauchy in the other. It is a uniform property ...
33 views

Proof verification of $x_n = \sum_{k=1}^n a_kq^k$ is Cauchy given $|a_k| \le C, |q| < 1, k\in\Bbb N$

Given a sequence $\{x_n\}$: $$x_n = \sum_{k=1}^n a_kq^k$$ and: $$\begin{cases} |a_k| \le C\\ |q| < 1\\ k\in\Bbb N \end{cases}$$ Prove $\{x_n\}$ is a fundamental sequence. By ...
65 views

Show that $\lim_{n\to\infty} \int_{0}^{1} f_n$ exists
Let $(X,d)= (C[0,1],d)$ where $C[0,1]$ is the set of real-valued continuous functions on $[0,1]$ and $d= \int_{0}^{1} |f-g|$ is the Riemann Integral. Suppose $(f_n)$ is a Cauchy sequence in $(X,d)$ ...