# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

1,544 questions
Filter by
Sorted by
Tagged with
1answer
372 views

3answers
1k views

1answer
170 views

### Convergence of Sequence of Real Numbers

Define a sequence of real numbers recursively as follows. Let $a_1 = 1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$. First, show the sequence is not monotonic. Second, show that $a_n \geq 1$ for all $n$ and ...
2answers
73 views

### Prove that $\left(a_{n}\right)_{n=1}^{\infty}$ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $0<q<1$

I'm stuck on a homework question, and could really use some help. Here is said question: "Assume that for every $n$ the following occurs: $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ when $0<q<1$ ...
1answer
145 views

1answer
151 views

### Completeness of metric spaces

edit: Another question about formatting. My questions on math.stackexchange keep cutting off the last few lines of my post. How do I fix this? The remaining lines show up in my edit box but not in the ...
2answers
761 views

2answers
187 views

2answers
73 views

0answers
2k views

1answer
248 views

### Revisiting a problem on Cauchy sequences

I posted earlier this week Proving a sequence is Cauchy given some qualities about the sequence I believe I solved the question myself, but my roommate has a different method of solving it, and is as ...