I have to show that for every $x \in \mathbb{R}$ and every $n \in \mathbb{N}$ $$|\sin(nx)| \le n|\sin(x)|  $$ In the previous exercise, I have showed that $|\sin(x)|≤|x|$ with the use of the mean value theorem. I think that I cannot use this approach this time. I also tried to write $\sin(nx)$ as a series expansion but that doesn't work either.     Does anyone know how I can solve this?
Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help.
The Bash Manual says   { list; }        Placing a list of commands between curly braces causes the list to be executed in the current shell context. No subshell is   created. The semicolon (or newline) following list is required.      The braces are reserved words, so they must be separated from the list   by blanks or other shell metacharacters.       The parentheses are operators, and are recognized as separate tokens   by the shell even if they are not separated from the list by   whitespace.   If I remove the semicolon, like this:  $ { date } &gt;    Why does it expect stdin input? A metacharacter is a character that separates words.  The semicolon and whitespace are both "shell metacharacters". Why can't whitespace separate words date and }? Why do we need a semicolon instead of a whitespace for separating the words? How best to determine if an argument is not sent to the JavaScript function How do I prove this using question using Cauchy-Schwarz inequality? How to round a number to n decimal places in Java What I would like is a method to convert a double to a string which rounds using the half-up method - i.e. if the decimal to be rounded is 5, it always rounds up to the next number. This is the standard method of rounding most people expect in most situations.  I also would like only significant digits to be displayed - i.e. there should not be any trailing zeroes.  I know one method of doing this is to use the String.format method:  String.format("%.5g%n", 0.912385);   returns:  0.91239   which is great, however it always displays numbers with 5 decimal places even if they are not significant:   String.format("%.5g%n", 0.912300);   returns:  0.91230   Another method is to use the DecimalFormatter:  DecimalFormat df = new DecimalFormat("#.#####"); df.format(0.912385);   returns:  0.91238   However as you can see this uses half-even rounding. That is it will round down if the previous digit is even. 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Reason I ask is I've had the unfortunate experience of my SD Card dying on me, put in a temporary SD Card and allowed Google to do the auto-restore.  Some things that were restored:   Saved wireless networks Background Some apps - strangely enough only a subset of my apps were restored (some with data, some without). All previous apps were installed via Android Market.   Then finally got my new SD Card, put that in and the following were restored   Saved wireless networks Some apps - again, same problem... only a subset (different subset, even fewer apps). This time no data seemed to have been restored either.   I think it would be very helpful to be able to view/manage a list of what is backed up, and thus would be restored (but I have yet to find this list... if it even exists). if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$ Let $P$ be a probability measure on $(\Omega, \mathscr{F})$, If $a,b,\in\mathscr{F}$ and $A\subset B$, then $P(B)-P(A)=P(B\setminus A)\ge 0$.  So far I have:  $A\subset B \implies B\setminus A = B\cap A^c = \emptyset$ as $A$ and $B^c$ are disjoint. Since $A\cup (B\setminus A)=A\cup (B\cap A^c)=(A\cup B)\cap (A\cup A^c)=B\cap\Omega = B$  Then for $A,B\in \mathscr{F}\implies P(A\cup (B\setminus A))=P(A)+P(B\setminus A) = P(B)$ by countable additivity.  Rearranging, $P(B)-P(A)=P(B\setminus A) \ge 0$. Suppose I have two tables that don't have many columns (3, for example). How can I position them side by side? How can I request a random row (or as close to truly random as is possible) in pure SQL? How can I make extra money with photography? I have photos in Getty Images, in Shutterstock, Imagekind and I recently added some to RedBubble, but I'm not having the expected results.  Does anyone have some history of success selling photographs or working as a photographer?  Examples of my work are available in .    Also asked by   I am an amateur who sometimes takes pictures of nature or random things. I've been told by several people that my pictures are worth money.  How can I monetize my best shots?  I don't really want to do a lot of work, editing, SEO or whatever. I'm looking for an easy way where I would just upload my pictures somewhere and let it go.  Maybe stock photos? People buying prints? I have no idea how that works...  EDIT: maybe a link to see which type of photography I do can help What are the current travel restrictions on individuals entering the United States from regions affected by Covid-19? This community wiki explains the current restrictions by the United States of America on the entry of individuals who may pose a risk of transmission of the SARS-Cov2, the virus which causes the Covid-19 disease. How do I make animations a steady speed? When I make an animation, it always starts out slow, then speeds up, then slows down at the end.  I cannot figure out how to fix this. How do you add a timer to a C# console application Just this - How do you add a timer to a C# console application? It would be great if you could supply some example coding. t-test for partially paired and partially unpaired data An investigator wishes to produce a combined analysis of several datasets.  In some datasets there are paired observations for treatment A and B.  In others there are unpaired A and/or B data.  I am looking for a reference for an adaptation of the t-test, or for a likelihood ratio test, for such partially paired data.  I am willing to (for now) to assume normality with equal variance and that the population means for A are the same for each study (and likewise for B). Use of Indefinite Articles(a,an) Consider the following sentences:  I am staying in a hotel nearby. I am staying in an hotel nearby.  I know that both are correct. But I would like to know why? I do not see any proper reason for the second one as the 'h' is not silent as in French. So, what could be the reason? Short story with competitors complaining to government about a new very popular synthetic food Synthetic foods are all the rage, and while some are new and novel, many are attempts to recreate the taste and texture of a natural foods, with varying degrees of success. One company introduces a new synthetic food that becomes wildly popular. The competition analyzes it, and based on discovering what natural creature it is based on, they petition the government to ban it.      They discover that the product is synthetic human flesh, so consuming it is not technically cannibalism, but it is similar, as they explain to the panel of government officials. How does sudo handle $HOME differently since 19.10?