real analysis – Help with a limit involving incomplete beta integral

In trying to prove that the limit of a certain function approaches 1 as the positive integer parameter $n$ approaches infinity, I have ended up with the following intermediate expressions:
$$f(n)=2^{1+2n}B_{1/2}(n,n+2)$$
$$g(n)=4^nB_{1/2}(n+1,n)$$
$$ h(n)=n(n-1)/2 left( frac{f(n)}{n+1}-frac{g(n)}{n-1}right)$$
Can somebody kindly help me with the evaluation of $ lim_{n to infty} h(n)$?
If somebody could also plug it in Mathematica, I would be highly obliged. Thanks for any help in advance.
P.S.: In the above the notation $B_z(a,b)$ stands for the incomplete beta function defined by:
$$B_z(a,b)=intlimits_0^z u^{a-1}(1-u)^{b-1} mathrm{d}u.$$

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Is it a good practice to hit actual database for Integrations test(Not prod but pre prod DB) for Jenkins Night builds?
The reason I am asking this is because our DAO uses procedures to perform simple CRUD operations so it is almost impossible to test operation like INSERTS/UPDATES/DELETE even against in memory H2 database.

real analysis – Open and closed sets in $mathbb{Q}$

Let $E$ be all $p in mathbb{Q}$ such that $2<p^2<3$. Could someone quickly check the following?

$E$ is closed in $mathbb{Q}$ because $E^c$ is open: consider $x in E^c$ so $x^2 geq 3$ or $x^2 leq 2$, and $x > sqrt{3}$ or $x<sqrt{2}$. If $x > sqrt{3}$ then by the density of the rationals in $mathbb{R}$ there exists $x>z>sqrt{3}$ and with $r=|x-z|$, we have $N_r(x) subset E$ so $x$ is an interior point and $E^c$ is open so $E$ is closed. If $x<sqrt{2}$ then by the density of the rationals in $mathbb{R}$ there exists $x<z<sqrt{2}$ and with $r=|x-z|$, we have $N_r(x) subset E$ so $x$ is an interior point and $E^c$ is open so $E$ is closed.

$E$ is also open in $mathbb{Q}$ because for some $x in E$, we know $2<x^2 < 3$ so $sqrt{2}<x < sqrt{3}$ and then similarly find $y$ and $z$ such that $sqrt{2} < y < x < z< sqrt{3}$ and take $r = min{y,z}$ and $N_r(x) subset E$. So $E$ is open.

We know compact subsets of metric spaces are closed so then since $E$ is open it is not compact.

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real analysis – Prove that the outer measure of an arbitrary sum of disjoint intervals is the infinite sum of the lengths of the intervals

The following is my HW Exercise:

Prove that if $I_1, I_2, dots$ is a disjoint sequence of open intervals, then $$left|bigcup_{k=1}^infty I_kright| = sum_{k=1}^infty mathscr{l}(I_k)$$ where $|cdot|$ denotes outer measure.

I found another, presumably simpler exercise:

Suppose $a, b ,c ,d$ are real numbers with $a<b$ and $c<d$. Prove that
$$|(a, b)cup(c, d)| = (b-a)+(d-c)iff (a, b)cap (c, d) = emptyset$$

I tried proving the simpler (second) exercise first as a stepping stone to proving the first exercise, but I didn’t get anywhere. Can someone please provide some hints on how the second exercise can be proven and perhaps also how the result of the second exercise can be used to prove the first exercise? Also, I haven’t learned about measurable sets yet because I am just getting started in Measure Theory.

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php – Real time post display on home page (as soon as they are published)

I have noticed this becoming a trend with comments and now posts. I’m sure many sites (especially community ones) will implement this.

Here’s exactly what I mean, click here: https://stackexchange.com/questions?tab=realtime

Posts appear in real time as soon as they are published.

There are also another version of this where the expand link appears with (1 question with the new activity). You can see this in action here if you wait for several seconds: https://superuser.com/

I’m wondering is there any way to do this on WordPress? With my loop?

Right now my loop looks like this:

<?php 
// Limit query posts to the current day
// 
$period = $_GET("period");
$day = $_GET("d");
$year = $_GET("y"); 
$args = array(
    'year' => (int) date($year),    
    'monthnum' => (int) date( 'M' ),    
    'day' => (int) date($day), 
    'date_query' => array(
        array(
            'after' => ($period), 
        )
    )
    
);

$query = new WP_Query( $args );

// The Loop
while ( $query->have_posts() ) :
    $query->the_post();
        
get_template_part( 'template-parts/content', get_post_type() );
            
endwhile;
the_posts_navigation();

?>

I think this would help a lot of people. Thanks.

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inequality – IF $a,b,c,d,e$ : Real number other than $-2≤a≤b≤c≤d≤e≤2$,Prove the following equation $frac{1}{b-a}+frac{1}{c-b}+frac{1}{d-c}+frac{1}{e-d}≥4$

Question
IF $a,b,c,d,e$ : Real number other than $-2≤a≤b≤c≤d≤e≤2$,Prove the following equation
$$frac{1}{b-a}+frac{1}{c-b}+frac{1}{d-c}+frac{1}{e-d}≥4$$

I tried to use Cauchy-Schwarz like
$$(frac{1}{b-a}+frac{1}{c-b}+frac{1}{d-c}+frac{1}{e-d})(b-a+c-b+d-c+e-d)≥16$$
I think I should prove it using -2≤a≤b≤c≤d≤e≤2, but I’m confused about how to do it. Can anyone help me?