There are ways of creating monotone sequences out of any sequence, and in this fashion we get the so-called limit superior and limit inferior. These limits always exist for bounded sequences. If a sequence \(\{ x_n \}\) is bounded, then the set \(\{ x_k : k \in \N \}\) is bounded.

So the monotonic convergence theorem simply states that: $\{s_n\}$ is a monotonic sequence. Then $\{s_n\}$ is convergent if and only if $\{s_n\}$ is bounded. by def this means $|s_n - L| < \epsilon$.Theorem (Bolzano-Weierstrass). Any bounded sequence (x n) of real numbers has a convergent subsequence. Proof. By the lemma, the sequence (x n) has a monotone subsequence (x n k). Since the original sequence is bounded, so is this subsequence. Thus this subsequence converges since monotone and bounded sequences always converge. Corollary.

Monotonic and bounded sequences throughout mathematics [closed] Ask Question Asked 1 year, 4 months ago. Active 1 year, 3 months ago. ... Using the principle "every bounded non-increasing sequence has an infimum", we can show that $(f_n)_n$ converges point-wise on $[0,1]$.Hausdorff (1) showed that for every totally monotone sequence Co, Ci, c2, • • there exists (essentially uniquely) a monotone nondecreasing real function <p(u), O^M^jl, such that n = 0, 1, 2, • • . Conversely, if 4>{u) is a monotone nondecreasing bounded real function on the interval O^w^l, then Amc„ =/„'(! monotone. Deﬁnitions. • A sequence (s n) is convergent iﬀ • A sequence (s n) is bounded iﬀ • A sequence (s n) is increasing iﬀ • A sequence (s n) is decreasing iﬀ • A sequence (s n) is monotone iﬀ (s n) is increasing or decreasing. We know some theorems relating these properties: If a sequence is , then it is . Monotonic, Upper bound and lower bound. Given any sequence {a n } we have the following terminology: 1. We call or denote the sequence increasing if a n < a n+1 for every n. 2. We call or denote the sequence decreasing if a n > a n+1 for every n. 3. If {a n } is an increasing sequence or {a n } is a decreasing sequence we denote it monotonic. 4.Earlier, we also saw that although convergent sequences are bounded, the converse is not necessarily true. Here, we prove that if a bounded sequence is monotone, then it is convergent. Moreover, a monotone sequence converges only when it is bounded. Theorem 9 (Monotone Convergence) A monotone sequence is convergent if and only if it is bounded.In constructive mathematics we do not need the axiom of choice to show that the (two-sided) Dedekind cuts form an ordered archimedean field which is complete in the sense that every Cauchy sequence converges.. However, it is possible to define a monotone bounded sequence, known as a Specker sequence, such that it is not provable constructively that the sequence has a limit.7.5 Bounded sequences converge (the other case) Corollary. If (a n) is bounded below and monotone non-increasing, then a n tends to the inﬁmum of {a n: n ∈ N}. 7.6 An example: 2cos π 2n+1 Let a 1 = √ 2 and for all n > 0 let a n+1 = 2+a nEvery bounded monotone sequence converges. Proof. We will prove the theorem for increasing sequences. The case of decreasing sequences is left to Exercise 4.3. So, let a sequence (a n) n≥1 increase. By the assumption of the theorem, (a n) n≥1 is bounded, that is, there exists C ∈ R such that |aBounded monotonic sequences We know that not every bounded sequence is convergent [for instance, the sequence an = (–1)n satisfies –1 an 1 but is divergent,] and not every monotonic sequence is convergent (an = n ). But if a sequence is both bounded and monotonic, then it must be convergent. Monotonic Sequence Theorem Every bounded, monotonic sequence is convergent. Deﬁnition A sequence {a n} is bounded above if there is a number M such that a n ≤ M for all n ≥ 1. It is bounded below if there is a number m such that m ≤ a n for all n ≥ 1. If it is bounded