These are just the examples from my textbook, but I don't think it did not explain well.  One of the problem was to prove the inequality   $$n&lt;2^n$$  for all integers $n$.  I understand we assume if $P(n)$ is true, $P(k+1)$ is true.  Thus $k+1&lt; 2^{k+1}$  But in the text book, they added one more line that I don't understand:  $$k+1&lt;2^k+1 \le 2^k+2^k=2\cdot2^k=2^{k+1}$$  I understand  $$k+1&lt;2^{k+1}$$  But how do I know $2^k+1$ is larger than $k+1$, and smaller than $2^{k+1}$?  I have one another example just in case,  proving $2^n&lt;n!$ using induction.  $$2^{k+1}=2\cdot 2^k&lt;2\cdot k!&lt;(k+1)k!=(k+1)!$$  I understand we should start with $2^{k+1}&lt;(k+1)!$, but  Where did $2\cdot k!$ come from? Textbook says it came from inductive hypothesis, but I still don't get it.  Please advise.
I'm having difficulty solving an exercise in my course.  The question is:      Prove that $n!\geq 2^n$.   We have to do this by induction. I started like this:   The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$.   Now proof for $(n+1)$ which brings me to: $(n+1)! \ge 2^{(n+1)}$  I think I can rewrite it somehow like this:  $$ {(n+1)} \times {n!}  \stackrel{\text{(definition of factorial)}}{\ge} 2^n \times 2 $$  $$ (n+1) \times 2^n \ge 2^n \times 2 $$  Then I think I can eliminate the $2^n$ and have something like this: $n+1 \ge 2$, or $n \ge 1$.  But I think I'm wrong here some where and was hoping somebody has some advice on this. How can I prove the above assumption?  Any help would be appreciated, kind regards.
Can you take the same feat multiple times? I own a 2016 Macbook (with the USB-C charging) - at the time and in response to all of the reports of poor cables and poor chargers, I bought a meter so I could see what the throughput of my cables/chargers was. However, having run some tests I'm now not sure what to consider normal / what is harmless to the battery.   For all tests I used an original Apple C-C cable - though was able to replicate with two third party cables I own.  I have the original Apple block charging at 14.3 V / 1.99 A and a third party block that charges the same.  Another third party block which advertises USB PD and 3 A is currently charging at 20 V / 1.43 A  I've done some research online but I can't find any articles that indicate what a normal USB PD output is relative to the devices that connect to it and would appreciate any guidance / assistance.   TL;DR: Have Apple charger which charges at one voltage/amperage. Bought several others advertised at PD which are higher voltage, lower amperage. Not sure what is 'safe', or 'fast', or does least overall damage to battery. Reverse the ordering of words in a string When should you use struct and not class in C#? My conceptual model is that structs are used in times when the item is merely a collection of value types.  A way to logically hold them all together into a cohesive whole.  I came across these rules :   A struct should represent a single value. A struct should have a memory footprint less than 16 bytes. A struct should not be changed after creation.   Do these rules work? What does a struct mean semantically? Deleted object would be re-saved by cascade (remove deleted object from associations) How Does Kylo Ren not stop Chewbacca's blaster? How does Kylo Ren stop the blaster shot in the beginning when being shot at by Poe, only to be unaware of the coming blaster shot from Chewbacca, which seems much more predictable.   I am sure he knew Chewbacca, Rey and Finn where there as Chewbacca turns with the opening of the doors when Rey and Finn come in.  EDIT in regards to Richards Answer: I just want to expand a little more on question. What I don't understand is that Kylo Ren uses the force, so it is just not a matter him not hearing. When Poe shoots him from behind, he makes no sound, but yet he turns and stops the blaster and Poe. That has to be due to the power he commands in the Force.  And this is from Obi-Wan Kenobi:     "Well, the Force is what gives a Jedi his power. It's an energy field   created by all living things. It surrounds us and penetrates us; it   binds the galaxy together."―Obi-Wan Kenobi —   So it seems to me that it would be very hard for him even in a weakened state to not even notice that he was being shot at by virtue of the force being a part of Sith and Jedi's alike. Even if he was too weak to actually stop the blaster, the Force should have been able to warn him. Change line spacing inside a node in TikZ Why is the sum of two random variables a convolution? Remotely changing hard drive boot order, windows 7 Conditionally use 32/64 bit reference when building in Visual Studio How to get GNOME Shell "Applications overview" to show the full name of applications Is it possible to hide curves' guides in edit mode? Why does Ubuntu 14.04 LTS use a non-LTS kernel version? I only recently became aware that LTS wasn't just an Ubuntu thing - there's LTS kernel versions too. Yet the kernel version used (3.13) in Ubuntu 14.04 LTS is not an LTS kernel version. Is it considered unimportant whether or not a kernel version is LTS? Diagonal lines in table cell I need this table:    (each row and column should have the same height and length, last cell should be divided by diagonal line). I tried the following code:  \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{array} \usepackage{makecell} \newcolumntype{x}[1]{&gt;{\centering\let\newline\\\arraybackslash\hspace{0pt}}p{#1}} \begin{document} \setlength{\extrarowheight}{0.1cm} \begin{tabular}{|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|}\hline &amp;&amp;&amp;&amp;20\\ \hline &amp;&amp;&amp;&amp;30\\ \hline &amp;&amp;&amp;&amp;45\\ \hline 15&amp;12&amp;18&amp;50&amp;\diaghead(-3,2){\hskip \hsize}{$a_i$}{$b_j$}\\ \hline \end{tabular} \end{document}   but text in the last cell is displayed incorrectly and cells are not the same.     How can I change that? WCF - Faults / Exceptions versus Messages We're currently having a debate whether it's better to throw faults over a WCF channel, versus passing a message indicating the status or the response from a service.  Faults come with built-in support from WCF where by you can use the built-in error handlers and react accordingly. This, however, carries overhead as throwing exceptions in .NET can be quite costly.  Messages can contain the necessary information to determine what happened with your service call without the overhead of throwing an exception. It does however need several lines of repetitive code to analyze the message and determine actions following its contents.  We took a stab at creating a generic message object we could utilize in our services, and this is what we came up with:  public class ReturnItemDTO&lt;T&gt; {     [DataMember]     public bool Success { get; set; }      [DataMember]     public string ErrorMessage { get; set; }      [DataMember]     public T Item { get; set; } }   If all my service calls return this item, I can consistently check the "Success" property to determine if all went well. I then have an error message string in the event indicating something went wrong, and a generic item containing a Dto if needed.  The exception information will have to be logged away to a central logging service and not passed back from the service.  Thoughts? Comments? Ideas? Suggestions?  Some further clarification on my question  An issue I'm having with fault contracts is communicating business rules.  Like, if someone logs in, and their account is locked, how do I communicate that? Their login obviously fails, but it fails due to the reason "Account Locked".  So do I:  A) use a boolean, throw Fault with message account locked  B) return AuthenticatedDTO with relevant information Auto-implemented getters and setters vs. public fields I see a lot of example code for C# classes that does this:  public class Point {     public int x { get; set; }     public int y { get; set; } }   Or, in older code, the same with an explicit private backing value and without the new auto-implemented properties:  public class Point {     private int _x;     private int _y;      public int x {         get { return _x; }         set { _x = value; }     }      public int y {         get { return _y; }         set { _y = value; }     } }   My question is why. Is there any functional difference between doing the above and just making these members public fields, like below?  public class Point {     public int x;     public int y; }   To be clear, I understand the value of getters and setters when you need to do some translation of the underlying data. But in cases where you're just passing the values through, it seems needlessly verbose. How do the Treasure Goblins work? Every once in a while I'll come across the sneaky devils and they always flee.  What determines when they flee (time, distance, etc.) and do I get better loot by killing them faster?  Is there anything else I should know in order to get the most out of the encounter?  EDIT: What I mean is: Is the final drop better based on how effectively I defeat them? In reading about the  protocol I noticed how much its proofs-of-work apparently depend on  of the SHA-256 hash function. And so  presumably do many other applications.  How do researchers go about proving uniformity of proposed hash functions? Is it just an empirical matter of running many inputs through the function and observing the distribution of outputs, or can one formally prove the property? Where could one (non-crypto-expert) learn about how such proofs are constructed and look one up e.g. for the case of SHA-256? If I run history, I can see my latest executed commands.  But if I do tail -f $HISTFILE or tail -f ~/.bash_history, they do not get listed.  Does the file get locked, is there a temporary location or something similar? What is a race condition? When writing multithreaded applications, one of the most common problems experienced is race conditions.    My questions to the community are:  What is the race condition? How do you detect them? How do you handle them? Finally, how do you prevent them from occurring?