How can I prove that a sequence of real numbers always includes a decreasing or an increasing subsequence?
32580
Prove there exists a subsequence of the real numbers such that it is monotonically increasing or decreasing.
55826
Convergence of sequence of real numbers and (sub)subsequence
141895
Proving that a sequence is monotone
111449
Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.
1396
cardinality of all real sequences
86465
Prove that every unbounded sequence contains a monotone subsequence that diverges to inifnity.
123357
Real Analysis - A sequence that has no convergent subsequence
25674
Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
140424
Proof of Divergence for a Sequence
78127
Convergence of non increasing and non decreasing sequences to the same limit
143424
Sequence problem involving inequalities
10933
Let $(x_n)$ be a bounded but not convergent sequence. Prove that $(x_n)$ has two subsequences converging to different limits.
112773
With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges
52518
Infinite sequence of digits without consecutive repeating subsequenes
112663
Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity
95860
The limit of the difference of two consecutive sequence members is equal to $0$. Can we conclude that the sequence itself has a limit?
107311
Convergence of a sequence and some of its subsequences