Basis Case:  $2! = 2\times1 = 2$  $2^2 = 4&gt;2$  Inductive Hypothesis:  $k!&lt;k^k$  Induction Step:  $k!&lt;k^k$  $k!(k+1) &lt; k^k(k+1)$  $(k+1)! &lt; k^{k+1} + k^k$  I'm confused on where to go from here, or if what I was doing was even the right way to go. I need advice on using induction for this question and on inequalities in general.
Call $P(k): k!&lt;k^k$, for $k\geq 2$ Test it out with 2, and it's true ($2&lt;4$).   Assume that $P(k)$ is true for some $k\geq2$.  Then show that $P(k+1)$ is true.   $P(k+1): (k+1)!&lt;(k+1)^{k+1}$  Ministep: $(k+1)!=k!(k+1)$ Ministep: $(k+1)^{k+1}=(k+1)^{k}(k+1)$    $k!(k+1)&lt;(k+1)^{k}(k+1)$  Can I just pull out that (k+1) and call it a day? That will make $k!&lt;(k+1)^{k}$.   We're already assuming that $k!&lt;k^k$, for some integer $k\geq 2$, and I don't think I need to prove that $k^{k}&lt;(k+1)^{k}$. I mean, I guess I could go prove that as well, but let's assume for the moment that I don't have to. Since we're assuming the original $P(k)$ is true, I should be able to write $k!&lt;k^{k}&lt;(k+1)^{k}$ right? That would prove the $k+1$ case. Is this the correct way to go about things? It seems too easy, but makes sense.  Thanks for confirmation and/or pointing out mistakes!
Is your MAC address revealed when you hit up a website? So I had been literally trying to fix my windows not being able to boot at all for more than 24 hours straight till now. I almost read every single article and troubleshooting step on the web to find a solution but nothing seems to work. I will keep it short as much as I can.  I did all of the following and it was:  "Reset this PC" gave an error "The required drive partition is missing"  "Refresh this PC" gave an error "The drive where windows is installed is locked. Please unlock and try again"  "System restore" gave an error something I don't actually remember but its like "Choose your OS and then try again"  "Automatic Repair" just failed to fix it and reverted me back to advanced options.   After all of that, I did  bootrec /fixboot bootrec /fixmbr bootrec /scanos bootrec /rebuildbcd bcdboot d:\windows (D is where I have the windows installed and C is for system reserved) bcdboot d:\windows /s c: /f ALL   All of these command prompt lines failed giving multiple errors like:  For REBUILDBCD, it said "Identified total windows installations: 1" then clicked "Y" to add it to the list and then it said "The requested system device cannot be found"  For bcdboot d:\windows it said Failure when attempting to copy boot files"   I wiped the disk multiple times using diskpart and installed fresh copies of windows on it 3 times but each time the installation will finish stage 1 (at a very slow pace) then when it restarts to continue installation, it simply won't boot (Windows logo and a spinning dot circle below forever)  I gave up and tried a totally new HDD and installed windows to it, it booted so normal and fast actually so I made sure it was not an issue with the windows.  Now I am booted from the new HDD and reformatted the old HDD and working on data recovery using easeus and it is finding my stuff luckily. It is also noticeable that the old HDD is making some noise while it is reading now as I scan for lost files.  My ultimate question is, will this HDD ever be able to used again? Even as a storage drive not as a drive I boot from? Because it is being seen through the bios boot menu and I see it in the disk management with healthy status. Can this HDD be fixed or I'm better off stop using it?  Here is a screenshot of the health status through SMART using HD Tune Pro: Setting Boundaries around city in google maps How can I debug my JavaScript code? Would time speed up near a large body of negative mass relative to observers in micro gravity? Time tends to slow down near objects with large amounts of positive mass relative to observers in micro gravity.  Considering that negative mass is the opposite of normal mass and would time tend to speed up near a body of negative mass relative to observers in micro gravity? How many seconds count until bomb detonates in Counter Strike? In Counter-Strike, is there any count (in seconds/minutes) until the bomb explode when it's first set-up? And can the timing be modified? I have gridded depth points in CSV format that I'm trying to add other map data to, in additional columns. I've tried joining fields, hoping that I could link the Shapefile data to the csv, but when I try to display the csv by graduated symbols based on the new fields, it goes blank. I used Nathan's tips from , but imported the csv as csv since I want to add map info to that, rather than adding csv info to a map.  See this picture below for the sea bottom temperature &amp; salinity voronois by month, underneath the csv points (a sample subset), with the joins already made. I added columns on the csv for each of the fields i'm intending to append.    I also tried spatial joins as per  however it says the layers aren't the same CRS (I checked, they are), then gets to 15% and hangs.  I also also tried spatial join through  but it hit a python error. I can try this again and look to debug the problem if this is the right way to go.  In short: appending polygon data as fields on a points-grid csv: what's the best way in QGIS 2.0? "As a beginner, I am happy that I learned a new thing."  You will recognize this as your DoReMi-s.( at 4:52)  What is the meaning of the "AS" there and when do we use that?  In depth answers would be appreciated.  Similarly:     Do I have your word as a lawman not to shoot me down like a dog in the street?    Can you explain this sentence?   Can I understand it as a lawman wouldn't shoot down people on the street or wouldn't have his word or what? Removing nodes from graphs such that one is dependent on other - ZIO $2010$, P$1$ Greetings Community! The above problem you see is a combinatorics problem I could not solve. :( This problem is from ZIO $2010$, P$1$.     Here is what I did: Observe that every graph can be divided into "sub-groups" which may help in solving. We first remove the empty set. Then the singleton set. Consider the bigger but smaller than the sub-parts example:    The singleton sets are $\{4,5,6\}$ which are at the least position in the graph.   The dual sets are $\{2,4\}, \{3,5\}, \{3,6\}$, the bottom - two portions in the graph. The triplet sets are $\{2,4,5\}, \{4,5,6\}$ which we may manually count but for the bigger graphs in the sub-part, this may prove to be difficult and this is where I am stuck.   Any help would be appreciated. Thanks. I've been trying to find this darn series of books for ages, I read them back in early 2001-02 and can't really remember much of them. I think there were 10-ish books, it was an easy read but a long series. On early publications, the covers, were mainly yellow, with small pictures of whatever happens in that book.  The book began with two siblings, a brother and a sister, running in a forest. She conjures up what can only be remembered as a golem type creature. He then uses a high magic 'blast' and wipes out the golem, knackering himself. His sister laughs at him for being 'powerful' but expending himself too much on one big magic blast. The beginning I remember vividly, the rest, not so much.  The story continues as a conflict happens, I believe two worlds are involved? The brother and sister are split up, but this is where my memory becomes vague.  The mechanics of the world seems that there are gems, or stones, set in the backbone, which in turn give access to magic. When the person uses them, they can draw too much, which ends up exhausting them. The 'big bad evil king' kills their father, and destroys their farm early on in the story. Let $(x_n)$ be a sequence of real numbers.  Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the original sequence $(x_n)$ itself converges to $x$ .              Thanks for any help. I heard Andrew Ng (in a video I unfortunately can't find anymore) talk about how the understanding of local minima in deep learning problems has changed in the sense that they are now regarded as less problematic because in high-dimensional spaces (encountered in deep learning) critical points are more likely to be saddle points or plateaus rather than local minima.   I've seen papers (e.g. ) that discuss assumptions under which "every local minimum is a global minimum". These assumptions are all rather technical, but from what I understand they tend to impose a structure on the neural network that make it somewhat linear.  Is it a valid claim that, in deep learning (incl. nonlinear architectures), plateaus are more likely than local minima? And if so, is there a (possibly mathematical) intuition behind it?   Is there anything particular about deep learning and saddle points? Windows 10 upgrade led to grub rescue Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$ I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome. How do I challenge an audit failure? I've run through Both SO and Meta searches for an answer, as well as reviewing the , and I cannot find a canonical process to challenge an audit failure.  I can find discussions of the single-choice "I Understand" topic that appear unresolved, in that no solution presently exists.  The particular question in point is .  Despite the number of up-votes, I feel that this quite clearly falls into a common closure category: "Here's my problem; how do I write the program?"  Compare this to   I also failed review on the first link I posted above, .  This looks like a case of code review, rather than SO.  I don't want to get rid of audits; I've learned good things from some of my failures, especially at the start.  However, in the past few weeks, a majority of my failures have been highly questionable, and I've drawn more than one suspension from those that are outright wrong by current standards.  If we have a challenge mechanism, please point me in the right direction.  Otherwise, it's simply not worth the annoyance any more, and I'll drop back to merely comments and answers. I am under DDoS. What can I do? Considering the following formulae:  (i) $1+2+3+..+n = n(n+1)/2$  (ii) $1\cdot2+2\cdot3+3\cdot4+...+n(n+1) = n(n+1)(n+2)/3$  (iii) $1\cdot2\cdot3+2\cdot3\cdot4+...+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4$  Find and prove a 'closed formula' for the sum  $1\cdot2\cdot3\cdot...\cdot k + 2\cdot3\cdot4\cdot...\cdot(k+1) + ... + n(n+1)(n+2)\cdot...\cdot (k+n-1)$  generalizing the formulae above.  I have attempted to 'put' the first 3 formulae together but I am getting nowhere and wondered where to even start to finding a closed formula. Creating image with specific latitude/longitude positions using GDAL? I have an ASCII file with latitude, longitude, and data_val in the following format.  35-13.643782N, 080-57.190157W, 118.6 ...   I have a GeoTiff image file, and I can easily view it.  I want to place a "pin" (can be a dot/flag/star or whatever is easiest) on the image at the specific latitude/longitude position found in the ASCII file.  Here is what I've managed to do so far:  My source image looks like this:  Driver: GTiff/GeoTIFF Files: /tmp/Charlotte SEC 100.tif Size is 16867, 12358 Coordinate System is: PROJCS["Lambert Conformal Conic",     GEOGCS["NAD83",         DATUM["North_American_Datum_1983",             SPHEROID["GRS 1980",6378137,298.2572221010042,                 AUTHORITY["EPSG","7019"]],             AUTHORITY["EPSG","6269"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4269"]],     PROJECTION["Lambert_Conformal_Conic_2SP"],     PARAMETER["standard_parallel_1",38.66666666666666],     PARAMETER["standard_parallel_2",33.33333333333334],     PARAMETER["latitude_of_origin",34.11666666666667],     PARAMETER["central_meridian",-78.75],     PARAMETER["false_easting",0],     PARAMETER["false_northing",0],     UNIT["metre",1,         AUTHORITY["EPSG","9001"]]] Origin = (-365041.822331817995291,240536.419747152860509) Pixel Size = (42.334586069440391,-42.334898968590878) Metadata:   AREA_OR_POINT=Area   TIFFTAG_DATETIME=2016:06:24 12:46:45   TIFFTAG_RESOLUTIONUNIT=2 (pixels/inch)   TIFFTAG_SOFTWARE=Adobe Photoshop CS5 Windows   TIFFTAG_XRESOLUTION=300   TIFFTAG_YRESOLUTION=300 Image Structure Metadata:   COMPRESSION=LZW   INTERLEAVE=BAND Corner Coordinates: Upper Left  ( -365041.822,  240536.420) ( 82d48'55.43"W, 36d13' 4.92"N) Lower Left  ( -365041.822, -282638.262) ( 82d35'10.11"W, 31d30'17.00"N) Upper Right (  349015.641,  240536.420) ( 74d51'46.40"W, 36d13'26.16"N) Lower Right (  349015.641, -282638.262) ( 75d 4'55.60"W, 31d30'36.99"N) Center      (   -8013.091,  -21050.921) ( 78d50'12.11"W, 33d55'36.35"N) Band 1 Block=16867x1 Type=Byte, ColorInterp=Palette   Color Table (RGB with 256 entries)     0: 255,255,255,255 ...   Here is what I've managed to cobble together in Python:  from osgeo import gdal, osr  src_filename = '/tmp/Charlotte SEC 100.tif' dst_filename = '/tmp/foo.tiff'  # Opens source dataset src_ds = gdal.Open(src_filename) format = "GTiff" driver = gdal.GetDriverByName(format)  # Open destination dataset dst_ds = driver.CreateCopy(dst_filename, src_ds, 0)  # Specify raster location through geotransform array # (upperleftx, scalex, skewx, upperlefty, skewy, scaley) # Scale = size of one pixel in units of raster projection # this example below assumes 100x100 gt = [-365041.822, 100, 0, 240536.420, 0, -100]  # Set location dst_ds.SetGeoTransform(gt)  # Get raster projection epsg = 4269            # http://spatialreference.org/ref/sr-org/lambert_conformal_conic_2sp/ srs = osr.SpatialReference() srs.ImportFromEPSG(epsg) dest_wkt = srs.ExportToWkt()  # Set projection dst_ds.SetProjection(dest_wkt)  # Close files dst_ds = None src_ds = None   But, I can't quite figure out how to place a "red dot" at 35-13.643782N, 080-57.190157W  I'm having to learn some new details here (nomenclature about GIS). How to reset KDE desktop to default on Ubuntu 14.04? KDE and Gnome Desktops were looking awesome on my Ubuntu 14.04 LTS, today everything seemed ugly. Slim long font, arabic letters in Chromium/Firefox are barely readable no matter how big I made them on system font settings. How can I get back to my previous looks settings, without compromising other settings and tweaks?  Here are my versions:  kde4-config -v:  Qt: 4.8.6 KDE Development Platform: 4.13.3 kde4-config: 1.0   kded4 --version:  Qt: 4.8.6 KDE Development Platform: 4.13.3 KDE Daemon: 4.13.3 What features of Mavericks are beneficial to you? This question serves to share and collect the enhancements that make a large difference to how you use your Mac.   Please post one feature per answer. Please also check to see if your answer has already been posted - duplicate answers will be deleted. To search answers for this question use  (directly from the question page) in addition to your search terms in the search box in the upper right hand corner of this page.  The best answers will not only list a feature, but provide details on how to configure that feature, and provide an image of how to use the feature to be more efficient or effective with Mavericks.