Let $(x_n)$ be a sequence that converges to a number x. Show that $(s_n)$ converges to x as well.
36145
A theorem about Cesàro mean, related to Stolz-Cesàro theorem
67638
Show that $x_{n+1}=x_n(2-ax_n)$ converges and find the limit
54659
How prove that $(a_{n})_{n \in \mathbb{N}}$ is convergent with $a_{0}=1$, $a_{n+1}= \sqrt{2a_{n}}$?
107735
What does the following sequence converge to?
133886
Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
60024
Proof that a sequence is convergent
49507
Is this sequence convergent?
86970
Let $\{x_n\}$ be a sequence in a metric space such that $\{x_{2n}\}$, $\{x_{2n+1}\}$ and $\{x_{3n}\}$ all convergent. Show that $\{x_{n}\}$ converges
79266
Proving a sequence defined by a recurrence relation converges
18675
$\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$
112430
Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$
111449
Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.
23140
Given $a_n, b_n$ are both convergent sequences such that $a_n \leq b_n$ for all $n$, show $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.
64721
A Sequence converges to a limit $L \in \mathbb R$ if, and only if, an $N$-th tail of the sequences converges to $L$ for some $N \in \mathbb N$
48500
Proof of convergence of sequences
142
Showing the sequence converges to the square root
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.