Recently I stumbled across a, to me, rather strange idea. I was messing around with the proof of $0.999... = 1$, when I figured that what $0.999...$ means is that those are all nines. That way I came upon a weird idea. Say $a = 0.999...$, then $a = 1 - x$. Yet, how do we define what $x$ is? It should say $x = 0$, but my theory did not. If $0.999...$ is just an endless sequence of nines, then why can't we say $x$ just is an endless sequence of zeroes, ending with a 1, like $\frac{1}{10^\infty}$? If we take the equation $$n = 0.9$$ for example, then, what would $1 - n$ be? Yes indeed, $0.1$. Following that theory, can't we say that $$0.999... = 1 - \frac{1}{10^\infty}$$Now, if $0.999... = 1$ this would be impossible. Go figure. $$1 = 1 - \frac{1}{10^\infty}$$ then $$\frac{1}{10^\infty} = 0$$ but that is impossible, because we cannot say $$10^\infty * 0 = 1$$When I came to this point, I really got stuck, because, in my head everything I did was right, however, it is impossible. Can somebody please explain to me what mistakes I may have made, and enlighten me about what else I did wrong?  Thanks in advance. Sjoerd Dorrestijn.  EDT: I prefer $\frac{1}{10^\infty}$ to use as an indication of $0.000...01$, even though $10^\infty = \infty$ in some way, I just seem to find this more clear.  EDT2: Just to be clear, I read a proof that $0.999...9 = 1$ because it'd be $1 - 0.000...0 = 1 - 0 = 1$. What I tried to prove here is that it is not  equal to $1$, because otherwise maths would collapse. My question was whether I am right or wrong. Since the original statement uses infinity (an infinite amount of nines) I think it is a must to use infinity as well. So, the question is if either the original statement is false, or if I made a mistake somewhere.
I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
Is it possible to automatically "add" \left and \right commands in front of all brackets? The source code should be left intact, but LaTeX should interpret every pair of brackets as if they were preceded by such commands. This is because adding \left and \right commands is time consuming and makes source code longer and less readable. I considered defining new commands such as \lb and \rb for \left( and \right), but it doesn't seem very elegant to me: what if I also want to add square brackets for example?  Any ideas? Thanks. $3\times3$ linear system organization /boot full of kernels – what to delete? We have some Ubuntu 16.04 servers. unattended-upgrades are automatically enabled since 16.04 and /boot is a separate partition. Due to the automatic security updates, the boot partition is running out of space with new kernels. We can't reboot the systems (for availability reasons), so the machine is still using the penultimate kernel. Which kernels should I remove? All but the current, the oldest and the newest? Do you guys have some recommendations? I have also noticed that the newest kernel has the status &quot;Half Configured&quot;. This kernel would probably not work, so should I remove this one and use an older kernel? Output of dpkg -l | grep linux-image: ii  linux-image-4.4.0-21-generic --&gt; old kernel ii  linux-image-4.4.0-34-generic --&gt; current kernel ii  linux-image-4.4.0-36-generic --&gt; new kernel ii  linux-image-4.4.0-38-generic --&gt; new kernel ii  linux-image-4.4.0-42-generic --&gt; new kernel ii  linux-image-4.4.0-45-generic --&gt; new kernel ii  linux-image-4.4.0-47-generic --&gt; new kernel ii  linux-image-4.4.0-51-generic --&gt; new kernel ii  linux-image-4.4.0-53-generic --&gt; new kernel iF  linux-image-4.4.0-57-generic --&gt; new kernel I'm using Linux Mint. My login shell (cat /etc/passwd | grep myUserName) is bash.  After I start my graphical desktop environment and run a terminal emulator from it, I can see that .bash_profile is not sourced (environment vars that are exported in it are unset). But if I log in from a text console (ctrl+alt+F1) or manually run bash -l from the terminal emulator, .bash_profile works fine.  Am I wrong when I think that .bash_profile should be sourced when X starts and all export'ed vars should be available in the terminal, running from X?  P.S. Placing everything in .bashrc and sourcing it from .bash_profile is not good idea (): environment stuff should be sourced only once. Beamer Tikz-cd Problem I have a training set with about 3000 positive instances and 3000 negative instances. But my test data set is pretty much un-balanced. The positive set only has 50 instances and negative has 1500 instances.This causes the precision very low. Are there any approaches to solve this problem? I use SVM to build classifier. Will using 'var' affect performance? Earlier I asked a question about  and got the answer that while it is only necessary for anonymous types, that it is used nonetheless to make writing code 'quicker'/easier and 'just because'.  Following  I saw that var gets compiled down to the correct type in the IL (you will see it about midway down article).  My question is how much more, if any, IL code does using the var keyword take, and would it be even close to having a measurable level on the performance of the code if it was used everywhere? The RSA signature operation is basically the same as encrypting with the private key.  In particular, both operations use the same kind of keys.  Is it safe to use the same RSA keypair both for encryption / decryption and for signing / verification? How do you find all $n$ such that $\phi(n)|n$ Idiomatic batch processing of text in Emacs? Assume $f:\mathbb R\rightarrow  {\mathbb R} $ is Lebesgue integrable, prove $\sum_{n\in\mathbb N } f(x+n)$ converges for almost everywhere x. This is a question discussed with friends. Thanks Input field value - remove leading zeros I have a textbox in Javascript. When I enter '0000.00' in the textbox, I want to know how to convert that to only having one leading zero, such as '0.00'. Bridged adapter in Virtual Box - cannot obtain IP I have virtual box with RHEL installed. My virtual box is running on windows 7 64 -bit. I configured bridged connection, but when I am restarting network interfaces, I receive following error:  Determining IP information for eth1... failed   On Windows XP it works without problems. I disabled Windows Firewall and installed latest Virtual Box, still without success.   NAT is working without any problems. What did I missed?  UPDATE  Ouput of ifconfig -a is:  eth0      Link encap:Ethernet  HWaddr 08:00:27:B6:FC:69           inet addr:10.0.2.15  Bcast:10.0.2.255  Mask:255.255.255.0           inet6 addr: fe80::a00:27ff:feb6:fc69/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:430 errors:0 dropped:0 overruns:0 frame:0           TX packets:346 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:36520 (35.6 KiB)  TX bytes:31455 (30.7 KiB)  eth1      Link encap:Ethernet  HWaddr 08:00:27:C3:84:4A           inet6 addr: fe80::a00:27ff:fec3:844a/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:1464 errors:0 dropped:0 overruns:0 frame:0           TX packets:17 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:135419 (132.2 KiB)  TX bytes:4230 (4.1 KiB)  lo        Link encap:Local Loopback           inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:16436  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)   Output of route -n  Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 10.0.2.0        0.0.0.0         255.255.255.0   U     0      0        0 eth0 169.254.0.0     0.0.0.0         255.255.0.0     U     1002   0        0 eth0 0.0.0.0         10.0.2.2        0.0.0.0         UG    0      0        0 eth0   Output of dhclient -v eth1 is:  Listening on LPF/eth1/08:00:27:c3:84:4a Sending on   LPF/eth1/08:00:27:c3:84:4a Sending on   Socket/fallback DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 6 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 9 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 15 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 18 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 13 (xid=0x6afb6c4c) No DHCPOFFERS received. No working leases in persistent database - sleeping.   Moreover the same machine with the same configuration works well on Windows XP. The problem is only on Windows 7. Will I die if I lose enough life to kill me from an effect that also heals me? What happens if I dip below 1 life from an effect that will also heal me after the damage is dealt?  For example:   If I bolt myself while I'm at 3 or less with a  on the board? Will I die while at 3 or less if I -2 , targeting myself? Will I die if a creature with my  on it damages me? Will I die if I target myself with enough damage to 'kill' me with a ?   and so on and so forth. You get the idea. I'm pretty sure there's a relatively simple answer that I just don't know about. Mysql resolving 127.0.0.1 to hostname Why does the /winsxs folder grow so large, and can it be made smaller? A Vista virtual machine I use only has a 10 GB virtual hard disk -- and I'm worried about it running out of space over time. I used  to check for the culprits..  And the primary culprit is that /winsxs or &quot;Windows Side-By-Side&quot; folder.  What is the Windows Side By Side folder? Well, that's .  All of the components in the operating system are found in the WinSxS folder – in fact we call this location the component store.  Each component has a unique name that includes the version, language, and processor architecture that it was built for.  The WinSxS folder is the only location that the component is found on the system, all other instances of the files that you see on the system are “projected” by hard linking from the component store.  Let me repeat that last point – there is only one instance (or full data copy) of each version of each file in the OS, and that instance is located in the WinSxS folder.   So looked at from that perspective, the WinSxS folder is really the entirety of the whole OS, referred to as a &quot;flat&quot; in down-level operating systems.  This also accounts for why you will no longer be prompted for media when running operations such as System File Checker (SFC), or when installing additional features and roles. That explains why the folder starts off big, but not why it gets larger over time – the answer to that question is servicing.   In previous versions of Windows the atomic unit of servicing was the file, in Windows Vista it’s the component.  When we update a particular binary we release a new version of the whole component, and that new version is stored alongside the original one in the component store.  The higher version of the component is projected onto the system, but the older version in the store isn’t touched.  The reason for that is the third part of why the component store gets so large.  Lots of other people :  A coworker recently received his every two year work laptop refresh and was in the middle of loading his new laptop with Vista. For some reason, his hard drive C: was partitioned to only 25GB and the rest of the space reserved for a much larger drive D:. Anyway, after loading many Windows Updates, he was left with hardly any drive space left on C:, and for some reason, the WinSxS directory had ballooned to over 8GB. While I knew what the WinSxS (Windows Side-by-Side) directory was for, I wasn't quite sure why that directory would explode in size so fast with normal operating system updates.  I don't really care about having all these duplicate files for safety, but I do care about the VM exploding due to an out of disk space error. So.. can /winsxs be safely pared down in any way? ..  Administrators should not, for any reason, take it upon themselves to clean out the /winsxs directory - doing so may prevent Windows Update and MSI from functioning properly afterwards.  Preventing accidental deletion from the directory is accomplished by putting a strong security descriptor on the directory that inherits to its children.  If this /winsxs folder gets any larger, I'm going to have to re-image this virtual machine.. which I really don't want to do! "Screaming" noise while steaming milk? Setting Zero values back to NULL Trying to hide spam/offensive deletions on user profiles announces them CDF of absolute value of difference in random variables