If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} - x_{n_{k-1}}\| < 1/2^k$
117179
Show that a Cauchy sequence has a fast-Cauchy subsequence
108696
If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence
3211
The sequence $x_{n+1}=\frac{x_n}{2}-\frac{2}{x_n}, x_0>0$ is bounded?
151226
Why is the sequence $x(n) = \log n$ **not** Cauchy?
73934
Subadditive sequences and the limit $\sup_{n\to\infty} a_n / n$
9789
Convergence of a sequence of non-negative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$.
60024
Proof that a sequence is convergent
135018
Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
2471
Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
31930
Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.
156905
Prove $f_n$ has a uniformly convergent subsequence on $[0,1]$.
69867
For each $\beta \in(0,1)$, give a sequence $\{a_j\}\subset(0,1)$ with $\prod (1-\alpha_j) = \beta $
141895
Proving that a sequence is monotone
124380
existence of sequence of polynomial
157447
Show that every monotonic increasing and bounded sequence is Cauchy.
48500
Proof of convergence of sequences
34071
Contractive condition implies the sequence is Cauchy