Articles of real analysis

Zeros of Fox-Write Function

This question is stimulate by the previous two question here and here. We are interested in studying the following special case of Fox-Write function \begin{align} \Psi_{1,1} \left[ \begin{array}{l} (1/k,2/k) \\ (1/2,1)\end{array} ; -x^2\right], x\in \mathbb{R}, k\in (1,\infty. \end{align} Where Fox-Write function $\Psi_{1,1} \left[ \begin{array}{l} (a,A) \\ (b,B)\end{array} ; z\right]$ is defined as \begin{align} \Psi_{1,1} \left[ […]

Find the limit of a recursive sequence

Let $(u_n)_n$ be a real sequence such that $$ u_{n+2}=\sqrt{u_{n+1}}+\sqrt{u_{n}},\,u_0>0,\,u_1>0. $$ Fisrt, it is easy to check that $(u_n)_n$ is well defined and $u_n>0$ for all $n\in\mathbb{N}$. The question now is show that $$ \exists p\in \mathbb{N}\,;\,\forall n\in\mathbb{N},\,n\geq p\implies u_n>1. $$ From this, we can deduce the limit of the sequence $(u_n)_n$.

Show that for $|f_n| \le g_n$ $\forall n$: $\lim_{n\to \infty} {\int_E g_n } = \int_E g \Rightarrow \lim_{n\to \infty} {\int_E f_n } = \int_E f$

Let $(f_n)_{n \in \Bbb N}$ be a series of measurable functions on E, that converges almost everywhere pointwise towards $f$. Let $(g_n)_{n \in \Bbb N}$ be a series of on $E$ integrable functions that converge almost everywhere on $E$ pointwise towards $g$. Also, $|f_n| \le g_n$ $\forall n \in \Bbb N$. I have to show […]

Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann Sum and Upper Riemann sum are bounded .More mathematically $m(b-a) \leq L(P,f) \leq U(P,f) \leq M(b-a)$ where $m,M$ are lower and upper bounds of the function $f$ respectively. Rudin […]

showing that $\lim_{x\to b^-} f(x)$ exists.

please I require help in showing that if $f$ is uniformly continuous on a bounded interval $(a,b)$, then $\lim_{x\to b^-} f(x)$ exists. edit: $f$ is uniformly continuous on $(a,b)$ implies that for every $\epsilon \gt 0$ there is a $\delta \gt 0 $ such that $|f(x) -f(y)| \lt \epsilon$ whenever $|x-y|\lt \delta$ for every $x,y$. […]

Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$

I need to prove this equivalence Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ if and only if $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$ I have already proved the reciprocal (the obvious part) but I am having difficulties with the other part Please any hint will be appreciated. Thanks in advance.

Proof of Divergence for a Sequence

Previous related question. After I negated the definition of a convergent sequence, I ended up with the following mathematical statement: $$\exists\ \epsilon > 0,\ \forall\ N \in \mathbb R\ \exists\ \mathbb N \ni n > N : |x_n – l| \ge \epsilon$$ Is this correct? I’d like clarification… Anyway, I’m now asked to use my […]

Property of limit inferior for continuous functions

I have the following question: Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X’$ (the dual space of the vector space X) we have that $(F(x_n))$ converge to $F(x)$ (that is: the sequence converge weakly on $X$). Let $F:X\longrightarrow\mathbb{R}$ a continuous function. Is it true that $\quad\displaystyle\liminf_{n\to\infty}|f(x_n)|\;{\color{red}\geq}\;{\color{red}|}f(x){\color{red}|}$? Recall that: […]

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ $\supseteq$ $I_3$ $\dots$ Now I used the nested interval property to say that there is at least a member […]

Prove $\sum_{n=1}^{\infty}f_n'(x)<\infty$ on $(0,1)$, when non-negative and increasing function $\lim_{x\to \infty}\sum_{n=1}^{\infty}f_n(x)<\infty$

When $f_n$ if non-negative and increasing on $(0,\ \infty)$ $$\lim_{x\to \infty}\sum_{n=1}^{\infty}f_n(x)<\infty$$ Prove that $$\sum_{n=1}^{\infty}f_n'(x)<\infty$$ on $(0,\ 1)$ a.e $[m]$. Is there the question means $f$ is differentiable? If so I will try mean value theorem. If not, I am totally stuck at the beginning, since $f$ is not mentioned absolutely continuous or f’ belong to […]