Articles of monotone functions

How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?

The formula is only applicable on values for $n\geq 2$. I know that the sequence is monotonic with a lower bound at $\frac 1 2$, but I am unsure how to find the supremum of the sequence. EDIT: $x_2 = \frac 1 2, x_3 = \frac 3 4, x_4 = \frac 5 8$. Does that […]

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, …$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable functions, i.e. for every $k\ge 1$ and every $x$ in $X$, $$0 \le f_k(x) \le f_{k+1}(x)$$. Next, set the pointwise limit of the sequence ${f_n}$ to be $f$. That is, […]

Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$

I have been looking at this for hours and it isn’t making anymore sense than it did in the first hour. If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit. ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$ What I though I had to […]

Expectation inequality for a set and its subset.

Given that a function $g(x)$ is a monotone increasing function. Its domain is an interval [c,d]. There are two sets $a_j \in A, j= 1,…,J$ and $b_k \in B,k=1,…,K$ on this interval, which satisfy $c_z \in C = B$\A and $c_z \geq a_j$ for all $a_j \in A$. For example, the interval $[c,d]=[0,8]$,$A=\{0,1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6,7\}$. […]

Why is a strictly monotonic mapping between intervals continuous?

Consider the following problem: Say $a,b\in\mathbb R$, $a <b$ and $f:[a,b]\rightarrow\mathbb R $ strictly monotonic, let’s say increasing for the sake of simplicity. Let $f ([a,b])=[c,d]$ for some $c,d\in\mathbb R$. Show that $f$ is continuous. This question is part of an old exam in introductory real analysis. Intuitively, I understand that a discontinuity in $f$ […]

Monotone functions and continuity

I want to show that if $f:(a,b)\to \mathbb R$ is a strictly monotone function and if $x_0\in (a,b)$ such that there exist two sequences $(a_n)$ and $(b_n)$ with $a_n<x_0<b_n$ and $\lim\limits_{n\to \infty}(f(b_n)-f(a_n))=0$, then $f$ is continuous at $x_0$. By monotonicity we can say that $\lim\limits_{n\to \infty}f(a_n)=f(x_0)=\lim\limits_{n\to \infty}f(b_n)$. Let $(x_n)$ be a sequence in $(a,b)$ such […]