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$f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly I was able to prove, hopefully correctly, that the sum converges uniformly. But, I'm not sure how to show it converges uniformly specifically to $f$.  The way I proved it converges uniformly was by using the Dirichlet convergence test, by taking $a_k(x) = \frac1n$ and $b_k(x) = f_k(x)$. $\sum_{k=1}^n{b_k(x)}$ is uniformly bounded on $[0,1]$ by $n+2$ for instance, and $a_n$ is monotone and uniformly convergent to 0 because it does not depend on the value of $x$.  I got rather stuck after this. I guess I have to use the continuity of the functions and the uniform convergence of both, but I can't figure out how to show that they converge to the same thing. Any help will be greatly appreciated. Examples of involutions on $\mathbb{R}$ Just recently I read up something about involutions (functions $f: A \rightarrow A$ such that $f(f(x))=x$, for all $x$ in the domain of $f$), and was wondering how many (if there is a small set of general functions) such involutions exist for $A = \mathbb{R}$, or maybe $A = \mathbb{R} - S$, where S is some set of points that would make the involution work if they were left out of $A$. In general I'm interested in real functions that are involutions, continuous or otherwise.  There were some examples I found , but any more would be certainly very interesting! My ceiling fixture has all of a sudden stopped working properly. Half the diodes are dark and the others are blinking. Oddly, the same exact ones on each strip work and don't work. I have included a picture to show which ones are blinking and which ones don't work. I am wondering if it is something that I can fix? Maybe replace that driver? And if so, is there a good guide online to walk me through that process? Thanks in advance. Why does the beta distribution become U shaped when $\alpha$ and $\beta$ <1? In the Beta distribution (used to model Bernoulli probabilities), the $\alpha$ and $\beta$ parameters can be interpreted as the number of heads$+1$ and the number of tails$+1$ seen. So, if they were both $2$, it would lean towards the coin being fair and have a maximum at $0.5$. If they are both $20$, the distribution would become even surer we're dealing with a fair coin and peak even more at $p=0.5$.  What I don't get is its behavior when $\alpha$ and $\beta$ both become $&lt;1$.   In that case, it becomes U-shaped and the density peaks at $p=0$ and $p=1$. Meaning the coin is likely to be two-sided. I know there is an intuition for this since I think I had an idea about it a long time ago. However, I've been trying to recollect all day and can't piece it together. Does anyone have an intuition? How can something finite become infinite? How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe? Who made the decision for Gandalf's promotion from 'grey' to 'white'? In the first book of the LoTR, after his fight with the Balrog of Moria, Gandalf the Grey changed to .   My question is, who made the decision to 'promote' him from grey to white? Was it ? Or was it the ? Or was it the ? When to use 'by' instead of 'at' or 'on' to specify time/date? Why am I getting strange results when using a power meter to measure CFLs? UK Visa in an expired Nigerian passport - what to use? I am in Nigeria, my passport will expire in 4 months and I have a document to apply for a UK visiting Visa with that passport number.  Should I use it like that or go for another passport. If I choose to get another passport, how will I use it because of the passport number on the about to expired one in the document I want to use for Visa. Suppose you do bad in undergraduate school in say computer science. But you do very well in a masters program in computer science. If you want to apply to a PhD program in computer science, will the masters degree grades offset the undergraduate degree grades? JComboBox Selection Change Listener? Which Stack Exchange sites use MathJax? As the title says, I could use an exhaustive list of all Stack Exchange sites that use MathJax.  If the list included any site-specific customizations made to the MathJax configuration on each site, that would be even better. What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? How do I format a String in an email so Outlook will print the line breaks? I'm trying to send an email in Java but when I read the body of the email in Outlook, it's gotten rid of all my linebreaks. I'm putting \n at the ends of the lines but is there something special I need to do other than that? The receivers are always going to be using Outlook.   I found a page on microsoft.com that says there's a 'Remove line breaks' "feature" in Outlook so does this mean there's no solution to get around that other than un-checking that setting?  Thanks I have not done much proofs before this and need some guidance. I know that for a simple graph such as this :  node - node - node -node   Removing the first and last vertex will not disconnect the graph. So far, I can think of two cases: Where there are leaves, or nodes with degree of 1. Where there are no leaves, which means all nodes have degrees 2 or more.   In the first case, removing the leaf will not disconnect the graph. In the second case, removing any should not disconnect the graph. Problem is, I don't know how to formulate this into a good proof. The merits of a partitionless filesystem I ran into something a couple of weeks ago that I'd never seen before: A filesystem (ext3 I believe) installed to a storage device without a partition. In essence /dev/sdb was the entire filesystem. I know many filesystems can be extended into empty space, so doing this allows extending without dealing with LVM or some other kind of volume manager, but are there any other advantages for setting up storage this way?  The specific case I saw was as the ephemeral data volume for a number crunching server, the boot and root volumes were traditional partitions on a different storage device entirely. - Display Blank when Referencing Blank Cell in Excel 2010 Is there a way to capture the viewport in Blender the same way as it is on 3DS Max? I know that by default I can render the viewport and everithing that is in the viewport to be the same as in the render. But, when comparing 3DS Max to Blender, I have certain limitations. In Blender I can render the view-port with vertices/edges/faces to be shown, to be selected, but not the reflections, shadows and such.  Is there a way to render the viewport in Blender as it is in 3DS Max?  Here's how I can do it in 3DS Max:  Viewport look    Render result of the viewport    Here is how I can do it in Blender:  Viewport look:      Render result of the viewport Reprojecting MODIS-Sphere ESRI GRID file to WGS84 using QGIS/GDAL I have an ESRI GRID file projected in MODIS-Sphere (here's the dataset ) that I need to convert to a WGS84 GeoTIFF.  I tried Raster > Projections > Warp (Reproject) and got this nonsense:   I then thought I'd use the MODIS reprojection tool or HEG to convert it, but they only accept HDF files and using QGIS' warp or translate functions to convert this file to HDF throwns an error. This guy was trying to do the same thing in Arc, no joy:   I looked for clues in the metadata, and the file seems a little goofy; gdalinfo yields:  Driver: AIG/Arc/Info Binary Grid Files: seto_uf        seto_uf.aux        seto_uf/.aux.xml        seto_uf/.DS_Store        seto_uf/dblbnd.adf        seto_uf/hdr.adf        seto_uf/log        seto_uf/metadata.xml        seto_uf/sta.adf        seto_uf/vat.adf        seto_uf/w001001.adf        seto_uf/w001001.tif        seto_uf/w001001.tif.aux.xml        seto_uf/w001001x.adf Size is 7468, 2911 Coordinate System is `' Origin = (-17345341.010356999933720,8423784.764957003295422) Pixel Size = (5000.000000000000000,-5000.000000000000000) Corner Coordinates: Upper Left  (-17345341.010, 8423784.765)  Lower Left  (-17345341.010,-6131215.235)  Upper Right (19994658.990, 8423784.765)  Lower Right (19994658.990,-6131215.235)  Center      ( 1324658.990, 1146284.765)  Band 1 Block=256x4 Type=Byte, ColorInterp=Undefined   Description = Seto_UF   Min=0.000 Max=101.000    NoData Value=255   So it can't determine the coordinate system.  There's a .tif included in the directory, but gdalinfo on that shows:  Driver: GTiff/GeoTIFF Files: w001001.tif        w001001.tif.aux.xml Size is 7468, 2911 Coordinate System is: GEOGCS["Unspecified datum based upon the GRS 1980 Authalic Sphere",     DATUM["unknown",         SPHEROID["unnamed",6371007,1]],     PRIMEM["Greenwich",0],     UNIT["degree",0.0174532925199433],     AUTHORITY["EPSG","4047"]] Origin = (-17345341.010356999933720,8423784.764957003295422) Pixel Size = (5000.000000000000000,-5000.000000000000000) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=BAND Corner Coordinates: Upper Left  (-17345341.010, 8423784.765)  Lower Left  (-17345341.010,-6131215.235)  Upper Right (19994658.990, 8423784.765)  Lower Right (19994658.990,-6131215.235)  Center      ( 1324658.990, 1146284.765)    So it says the units are degrees but those huge corner coordinates and origin values don't agree.  Any ideas?  My setup: OSX 10.9.4 QGIS 2.6.1 Brighton GDAL 1.11.2 (apparently there are newer versions but this is the latest you can get on OSX without having to compile it yourself) Version error when uploading Change Set I’m trying to deploy custom objects, custom fields, page layouts, compact layouts, Lightning pages and Global value set (global picklist) , when uploading changeset it is giving me API version error as out production platform version 46.0 and SB version is 47.0 , I’m not sure how to bring the version down to 46.0 (There are no Apex triggers/classes in the changeset). Any help would be really appreciated.