I wanted to {$e^{ik}|k\in N$} Is dense in $S^1$ My Attempt:  On the contrary, assume that it is not dense in $S^1$. So there exists a point in $S^1$ which is not a limit point, say $e^it $ for some $t\in R$ That means $\exists \epsilon &gt;0$ such that $\forall k\in N ,|e^{ik}-e^{it}|&gt;\epsilon $ that is $|e^{i(k-t)}-1|&gt;\epsilon $ I was expecting some contradiction to occur. But I am not able to see. Any help will be appreciated
Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$?
The following happened playing Minesweeper:   How do I figure out which square to click? Lightning force:createRecord capture redirect after save Is there a way to use force:createRecord but capture the redirect to the newly created record page that occurs after saving the record? Finally, prevent the redirect from completing?  createRecord : function (component) {     var createRecordEvent = $A.get("e.force:createRecord");     createRecordEvent.setParams({         "entityApiName": "Event__c"     });     createRecordEvent.fire(); } What happens to matter in a standard model with zero Higgs VEV? Suppose you reset the parameters of the standard model so that the Higgs field average value is zero in the vacuum, what would happen to standard matter?  If the fundamental fermions go from a finite to a zero rest mass, I'm pretty sure that the electrons would fly away from nuclei at the speed of light, leaving positively charged nuclei trying to get away from each other.  Looking at the solution for the Hydrogen atom, I don't see how it would be possible to have atoms with zero rest mass electrons.  What happens to protons and neutrons?  Since only a very small part of the mass of protons and neutrons is the rest mass of the quarks, and since they're flying around in there at relativistic speeds already, and since the nuclear force is so much stronger than the electrical force with an incredible aversion to naked color, would protons and neutrons remain bound assemblages of quarks and virtual gluons?  Would they get a little larger?  A little less massive?  What would happen to nuclei?  Would they stay together?  If the protons and neutrons hold together and their properties change only some, then I might expect the same of nuclei.  Different stable isotopes, different sizes, and different masses, but I would expect there would still be nuclei.  Also the W and Z particles go to zero rest mass.  What does that do to the electroweak interactions?  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I was under the impression that we would only need to acquire Enterprise licences for the actual production servers, and we could get by with developer edition in our developer and staging environments because they are not "production".  Other sources have told me that we need to have an enterprise licence on all of the above mentioned machines, and that the developer edition is only meant for a single developer on their workstation.  Since developer edition has all the sexy features of Enterprise, I can't really see the value of it on a workstation class machine, especially for developing and testing the type of high availability system we are building.  If we have to fork out Enterprise licences for a dev server, that will just about kill our proof of concept stage, thus killing the project.  Forcing an enterprise licence on a staging environment will make management just want to skip staging altogether. 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This implies that there is work done, but what force is doing this work? I have the value 25.00 in a float, but when I print it on screen it is 25.0000000. How can I display the value with only two decimal places? What is / how can I get a student-safe power supply for electronics projects?  I'm looking for a student-safe power supply for solder less breadboard.  I'm developing a course to teach electrical engineering to grade school students as an after-school program.  I'm wrapping up the first course and I've learned a lot, including that I need to find a safer power supply.  We're mostly using 9V batteries and connecting them to the breadboard via alligator clips.  After lots of searching, I'm realizing that I need to turn towards the expert community for guidance.  I hope that with the rise of the Makers  this question will become increasingingly relevant with time as more kids get into electronics.  Inextricably linked to this question is what is the minimum power source to use.  I've liked using 9V batteries since we can connect to them using alligator clips; but I worry that they are just a little too powerful.  I can imagine them somehow starting a fire with them.  My #1 priority is to keep the students safe.  Kids just have no concept that they can cause damage to themselves and property.  When I ask "where is that smoke coming from?" I'll get a gleeful response from the student that is seeing how many batteries he could hook together and short out.   I really need to limit the damage they can do.  I mainly don't want anyone electrocuted or anything to catch fire.  Kids will routinely attach LEDs backwards, leave them like that and then act surprised when they explode.   After safety, I want to protect the equipment.  Kids will hook up a LED backwards all the time.  After protection, I want to give the kids feedback as to what is going on.  A light that indicates power is on would be awesome.  A light or siren or circuit break that indicated a short would be awesome too.  Battery powered supplies are slightly preferred, just to keep upfront costs low and to avoid having to run outlets to each table.  Boundaries:   We are only using simple LEDs, BJT transistors, resistors and maybe a timer IC.    Similiar Questions:       References:  Course details are here:   We currently sometimes use  from the Squishy Circuits Store and/or a 9V battery with alligator clips.  Epilogue  I found it tougher than expected to find a suitable wall wart, so I'm listing what I finally found for future reference.  I'm about to test the following combination:         Epilogue 2  A year-plus in, I've decided to drop the in-line power switch since it wasn't obvious when it was 'on' vs. 'off'.  I'm instead relying upon the students to unplug the rat-tail from the supply, which is quick and easy.  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