I'm looking for a proof that ensures that for every natural number $n &gt; 1$ there is always a prime $p$ such that $\sqrt{n} \leq p \leq n$ holds.  I know you can deduce my previous claim as a corollary of some famous results like "Bertrand's postulate", but I was wondering if it does exist any easy self-contained proof.
Prove that for all $n&gt;2$ there exists at least one prime $p$ such that $\sqrt{n}&lt;p&lt;n$ using elementary methods.  My try: If not, $\sum_{p&lt;\sqrt{n}} (\lfloor n/p \rfloor-\lfloor \sqrt{n}/p \rfloor) \geq n-\sqrt{n}$  then...?
While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? Is 20 watts of electricity dangerous? I have a few circumstances which invlove someone being shocked with 20 watts of electricity and whether it would be deadly. So here they are:   Would 1000 volts at 20 miliamperes of AC (2MHz frequency) be dangerous or deadly? Would 1000 volts at 20 miliamperes of DC be dangerous or deadly? Would 10 Kilo volts at 2 miliamperes of AC (2MHz frequency) be dangerous or deadly? and Would 10 Kilo volts at 2 miliamperes of DC be dangerous or deadly?   Thanks for everyones help. Why is saltatory conduction in myelinated axons faster than continuous conduction in unmyelinated axons? The system L0 is defined as follows:  Axioms: A1 (α → (β → α))  A2 2. (α → (β → γ) → ((α → β) → (α → γ))  A3 ((¬β → ¬α) → ((¬β → α) → β))  In one of my problem sheets, I am told that I am allowed to use the following theorem: if   ⊢ (α → α)    ⊢ (α → α) → (α → α)  My attempt to solve/prove the second one  ⊢ (α → α) → (α → α) choose 2 axioms which match.  α →α   β→α  γ→α  (α → (α → α) → ((α → α) → (α → α))  I'm struggling to see an obvious way of proving the theorem.  Is it provable? And if so, can you point me in the right direction?  Thanks a lot! Ampscript date range lookup I want to use LookupOrderedRows and return rows between a start and end date (last month range). What is the best way to handle using Ampscript?   I can add one date to the where using   "DateField", @varStartDate,   But if I add the below to add additional where filters  "DateField", @varStartDate,"DateField", @varEndDate,   I get this error:     "Duplicate Key Field Name: DateField"   EDIT: Also I tried the range as the var for the LookupOrderedRows using   set @strRange = StringToDate(Format(DateDiff(@strStartDate, @strEndDate, "M"),"MM/dd/yyyy"))  and then in the LookupOrderedRows just using  "DateField", @strRange  But that got an error as well. Stone-Weierstrass theorem with $p(1/x)$ I was trying to fit my data into various models and figured out that the fitdistr function from library MASS of R gives me Negative Binomial as the best-fit. Now from the  page, the definition is given as:     NegBin(r,p) distribution describes the probability of k failures and r   successes in k+r Bernoulli(p) trials   with success on the last trial.   Using R to perform model fitting gives me two parameters mean and dispersion parameter. I am not understanding how to interpret these because I cannot see these parameters on the wiki page. All I can see is the following formula:    where k is the number of observations and r=0...n. Now how do I relate these with the parameters given by R? The help file does not provide much information either.   Also, just to say a few words about my experiment: In a social experiment that I was conducting, I was trying to count the number of people each user contacted in a period of 10 days. The population size was 100 for the experiment.   Now, if the model fits the Negative Binomial, I can blindly say that it follows that distribution but I really want to understand the intuitive meaning behind this. What does it mean to say that the number of people contacted by my test subjects follows a negative binomial distribution? Can someone please help clarify this? Does C# have anything comparable to Python's list comprehensions? From my old HD (Ubuntu 16.04) I copied ~/.thunderbird and other programs to the new disk with Ubuntu 18.04 installed. All went good. I tried to install Opera. The installation was unsuccessful, a window asking something was held. After that Thunderbird Local Folders -> Inbox (only) was invaded by large icons like this screenshot  Flowers, plants, trucks and others. I restarted but those images are still there. Opera was successfully installed. What can I do to remove them? Select from list of reasons when performing downvotes How many ways can the average of n dice be a? What exactly is the difference between a parametric and non-parametric model? I am confused with the definition of non-parametric model after reading this link  and .   Originally I thought "parametric vs non-parametric" means if we have distribution assumptions on the model (similar to parametric or non-parametric hypothesis testing). But both of the resources claim "parametric vs non-parametric" can be determined by if number of parameters in the model is depending on number of rows in the data matrix.  For kernel density estimation (non-parametric) such a definition can be applied. But under this definition how can a neural network be a non-parametric model, as the number of parameters in the model is depending on the neural network structure and not on the number of rows in the data matrix?  What exactly is the difference between parametric and a non-parametric model? If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.  I'm wondering if you can help me with this one? In a multi server architecture, is it possible to find out from which machine a user logged into current machine using ssh?  For example, user1 logged into host1. From there, the user logged into host2 as sruser using ssh.  In above example, would it be possible to trace the sruser session of host2 back to user1 session in host1?  Would it be possible to get the hostname or terminal of user1 session? $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. How should an educational institute woo good faculty? Flag indicator present on Meta, but no flags visible Correct usage of "lest" Which of the followings is/are the correct usage of the word "lest"? How are these different from each other?   ...hesitant to speak out lest he be fired. ...hesitant to speak out lest he'd be fired. ...hesitant to speak out lest he would/will be fired.   P.S. I took point number 1 as the reference for this question from the definition of lest from the Merriam-Webster Dictionary. What is the word that describes the absence of bureaucracy? What is the word which describes the absence of formless, of long procedure with bureaucrats? I was thinking about formsless but it sounds weird... Determine resultant velocity of an elastic particle-particle collision in 3d space So I have two particles that have collided in 3 dimensional space. I want the particles to rebound off of each other in an elastic manner. How do I determine the resultant velocity vector if I know: Both particles' initial velocity vector, their masses, and their initial position vectors relative to the origin.  I tried to follow the instructions on  but I am not sure how to make their equations work without polar coordinates / how to translate my data to polar coordinates.  Also, if you answer my question in terms of polar coordinates, can you explain which angle is theta and which is phi? Since there are 2 different conventions, I am not really sure which angle is being referred to in the explanation on the aforementioned website.  EDIT: This isn't for a class. I'm writing a physics engine for a gravity simulation I am making using Unity. I haven't taken kinematics since I was in high school, which is why I am so rusty.