$\{x_n\}$ be a bounded sequence of distinct real numbers , $|x_{n+1}-x_n|<|x_n-x_{n-1}|,\forall n\in \mathbb N$ , then is $\{x_n\}$ convergent?
83790
Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
49507
Is this sequence convergent?
130877
Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
65993
Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?
117649
$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent
50087
Convergence of $a_{n+1}=\frac{1}{1+a_n}$
48500
Proof of convergence of sequences
119858
Proof: Every convergent sequence of real numbers is bounded
132713
The space $c_0$ of sequences that converge to 0 is not reflexive
10933
Let $(x_n)$ be a bounded but not convergent sequence. Prove that $(x_n)$ has two subsequences converging to different limits.
40797
$(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
9244
Does the sequence $\{\sin^n(n)\}$ converge?
111211
Does $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ imply that $a_n$ convergent?
47740
Let $x_1=a>0$ and $x_{n+1}=x_n+\frac{1}{x_n} \forall n\in \mathbb N$. Check whether the following sequence converges or diverges.
87834
How can we show that the following sequence converges?
16970
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
66198
Sufficient condition for convergence of a real sequence