# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

1,543 questions
Filter by
Sorted by
Tagged with
0answers
71 views

3answers
3k views

### Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
2answers
485 views

### Finite Cauchy Sequence

How do you prove that given a sequence that is finite, the sequence always converges? Or, must the sequence be cauchy for this finite sequence to converge? How can a cauchy sequence be finite? If a ...
2answers
691 views

### Sum of the distance between consecutive terms implies the sequence is Cauchy.

If a sequence $(x_n)_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum_{n\geq 1} ||x_n-x_{n+1}||<\infty$, show that it is Cauchy. This isn't a complete answer, but here's my train of thought. ...
1answer
56 views

### Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
3answers
71 views

### Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
3answers
2k views

### If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
2answers
3k views

### Cauchy convergence in probability implies the existence of a (finite a.e.) limit $X$

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an $X$ (finite a.e.), such that $X_n$ converges to $X$ in probability. The problem's hint suggests ...
3answers
930 views