${x_n}$ is a sequence such that $|x_{n+1} - x_n| \leq \frac{1}{2}|x_n - x_{n-1}|$. Prove $x_n$ is convergent.
63523
$|a_{n+1}-a_n| \leq \frac{1}{2} |a_{n}-a_{n-1}|$. Prove $(a_n)$ is Cauchy
87834
How can we show that the following sequence converges?
61731
Prove that for each $n \in \mathbb N$, $\gamma_n=1+\frac{1}{2}+\dots+\frac{1}{n}-\int_1^n \frac{1}{x}dx$ is convergent
9044
Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$
49507
Is this sequence convergent?
60024
Proof that a sequence is convergent
2471
Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
132140
Convergence of Sequence of Real Numbers
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
98762
$f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?
121774
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 $
107399
A sequence and convergence
40797
$(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
67453
If $x_{n}$ is decreasing and $\sum x_{n}$ converges, prove that $\lim nx_{n} = 0$
6815
$\lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c , c > 0$ . Then, is $\{x_n/n\}$ convergent?
54659
How prove that $(a_{n})_{n \in \mathbb{N}}$ is convergent with $a_{0}=1$, $a_{n+1}= \sqrt{2a_{n}}$?
67638
Show that $x_{n+1}=x_n(2-ax_n)$ converges and find the limit