Prove that $\lfloor \log_{10}(xy)\rfloor \geq \lfloor \log_{10}(x)\rfloor+ \lfloor \log_{10}(y)\rfloor$
A simple inequality for floor function
Do we need inertial frames in Lagrangian mechanics? Proof of uniform limit of Continuous Functions The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all. I'm consistently receiving significant delays on Stack Overflow when opening the flag dialog on questions. Delays do not occur on any other StackExchange site I have checked. This started a couple days ago (as far as I can tell).  Delays occur not only when loading the initial screen, but also when loading the submenus. This is slightly confusing for a couple seconds (until you're used to it) since it appears as if the flag dialog has simply vanished.   What's going on here? I demand to know, otherwise somebody will be very upset.  Not me. I'm not going to get that upset over something like this.  But someone will.    Specs (though I know of several others who have this issue):  Windows 7 64-bit Chrome Version 30.0.1599.69m     Ping: 31ms Download: 6.88mbps   Upload: 1.07mbps headed or heading, started or starting? Consider:  Two units are headed your way. Two units are heading your way. Don't get me started! Don't get me starting!  Is there any difference in meaning? And I don't see why passive voice is used here. I would think they have similar meaning here. Help me to clarify! I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning several categories and functors between them?   Mac Lane suggests reversing all arrows in all categories and leaving the functors invariant in his book "Categories for the working mathematician" (page 32), but it seems that Mac Lane himself doesn't always follow this discipline (when he states "the dual of Yoneda Theorem", he doesn't reverse the arrows in $\mathbf{Set}$ !) Can anyone give me a formal definition of "dualize a statement in category theory"? Some mesh-edges in object-mode aren't displayed So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. convert bash `ls` output to json array How to create a SQL Server function to "join" multiple rows from a subquery into a single delimited field? To illustrate, assume that I have two tables as follows:  VehicleID Name 1         Chuck 2         Larry  LocationID VehicleID City 1          1         New York 2          1         Seattle 3          1         Vancouver 4          2         Los Angeles 5          2         Houston   I want to write a query to return the following results:  VehicleID Name    Locations 1         Chuck   New York, Seattle, Vancouver 2         Larry   Los Angeles, Houston   I know that this can be done using server side cursors, ie:  DECLARE @VehicleID int DECLARE @VehicleName varchar(100) DECLARE @LocationCity varchar(100) DECLARE @Locations varchar(4000) DECLARE @Results TABLE (   VehicleID int   Name varchar(100)   Locations varchar(4000) )  DECLARE VehiclesCursor CURSOR FOR SELECT   [VehicleID] , [Name] FROM [Vehicles]  OPEN VehiclesCursor  FETCH NEXT FROM VehiclesCursor INTO   @VehicleID , @VehicleName WHILE @@FETCH_STATUS = 0 BEGIN    SET @Locations = ''    DECLARE LocationsCursor CURSOR FOR   SELECT     [City]   FROM [Locations]   WHERE [VehicleID] = @VehicleID    OPEN LocationsCursor    FETCH NEXT FROM LocationsCursor INTO     @LocationCity   WHILE @@FETCH_STATUS = 0   BEGIN     SET @Locations = @Locations + @LocationCity      FETCH NEXT FROM LocationsCursor INTO       @LocationCity   END   CLOSE LocationsCursor   DEALLOCATE LocationsCursor    INSERT INTO @Results (VehicleID, Name, Locations) SELECT @VehicleID, @Name, @Locations  END      CLOSE VehiclesCursor DEALLOCATE VehiclesCursor  SELECT * FROM @Results   However, as you can see, this requires a great deal of code.  What I would like is a generic function that would allow me to do something like this:  SELECT VehicleID      , Name      , JOIN(SELECT City FROM Locations WHERE VehicleID = Vehicles.VehicleID, ', ') AS Locations FROM Vehicles   Is this possible?  Or something similar? Is it possible to resize an image in an answer? For example,  has  that should have the image in the answer, but it's too big.  Is there a way to handle resizing? Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light (weak) homotopy equivalence Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic Intersection of two lists in Bash I am a UI designer and developer and have been designing websites for aound 10 years. I want to get into user experience design. I know how to do a design, but I am lagging the terminologies and the technical things in the user experience field.   So how do I go about getting the skills and mindset necessary for being a User Experience Designer? Why and how is the speed of light in vacuum constant, i.e., independent of reference frame? How do I properly compare strings in C? At the start of Harry Potter and the Philosopher's/Sorceror's Stone Voldemort's Killing Curse backfires      due to Lily's sacrifice protecting Harry from Voldemort.   leaving Harry with his scar      and a loose piece of Voldemort's soul   and Voldemort as a loose soul, self-described as "Less than a ghost".  Voldemort was not dead, but had no physical body (until he got better in book 4).  What happened to his body?  Did it disappear?  Or was it left in the rubble of the house, alongside the bodies of James and Lily? Override a method via ObjC Category and call the default implementation?