Showing that $x_{n+1} = 1/2(x_n + 2/x_n)$ is a decreasing sequence.
31174
Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
88782
Finding the limit of $x_1=0 , x_{n+1}=\frac{1}{1+x_n}$
142509
If $\alpha\in (0,1]$ and $x_n=\alpha x_{n-1}+(1-\alpha)x_{n-2}$, show that the sequence $\{x_n\}$ is convergent.
101766
Recursive sequence convergence.: $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$, where $s_1>s_2>0$
35091
How can I prove this sequence is decreasing?
135018
Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
27217
For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$
34723
Proof of a limit for a recursively-defined sequence
152764
Let ($x_n$) be a monotone sequence and contain a convergent subsequence. Prove that ($x_n$) is convergent.
64885
Irrational and rational sequence proof
82883
Show that the sequence $a_{n+1}=a_{n}+\dfrac{a^2_{n}}{n^2}$ is upper bounded
95000
Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
86965
Limit of this recursive sequence: $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$
141895
Proving that a sequence is monotone
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.
93545
Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$
133451
To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$