I have the following problem: Let $|x_{n+1} - x_n| < 1/3^n$. Show that $(x_n)$ is a Cauchy sequence.
70746
Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.
142928
Proving $\left\{ \frac{1}{n} \right\}$ is Cauchy
55464
Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$
104033
Confused with proof that all Cauchy sequences of real numbers converge.
107636
verify, that this sequence is cauchy-sequence
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 $0$
54982
How to prove that $x_n=1+\frac{1}{2!}+\cdots+\frac{1}{n!}$ is a Cauchy sequence?
143545
If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.
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.
85217
Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$
82883
Show that the sequence $a_{n+1}=a_{n}+\dfrac{a^2_{n}}{n^2}$ is upper bounded
62439
If $x_1 = 3$, $x_{n+1} = \frac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is bounded below by $0$, above by $4$.
98762
$f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?
26837
Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
7507
Proof that $\lim{a_n}=L$ when $n$ goes to infinity, then $\{a_n\}$ its a Cauchy Sequence
26968
Proof that the series expansion for exp(1) is a Cauchy sequence