$\{{x_n}\}_{n \geq 1}$ be a sequence such that $|x_{n+1} - x_n| \leq \frac{1}{2} |x_n - x_{n-1}|.$ Prove ${x_n}$ is convergent
54427
$\left|x_{n+1} - x_n\right| < C\left|x_n - x_{n-1}\right|$ prove this is Cauchy
30488
Convergence of a sequence implies convergence of the norm of the sequence
23140
Given $a_n, b_n$ are both convergent sequences such that $a_n \leq b_n$ for all $n$, show $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.
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 $x_1 >0$
99933
Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$
95515
Is $x_{n+1}=\cos x_n$ convergent?
95095
$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.
111353
Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
147841
Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$
87834
How can we show that the following sequence converges?
42300
Let the sequence be defined recursively $X_{n+1}=pX_{n}+q$ for p nonzero and $X_{1}$ arbitrary. In what conditions does the sequence converge?
145863
sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
63562
Show that $x_{n+1}-\dfrac{x_n}{2}$ converges to zero implies that $(x_n)_n$ also converges to zero.
837
Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.
79266
Proving a sequence defined by a recurrence relation converges
35731
Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$.
153121
show that $\{x_n\}$ is a cauchy sequence