I was solving this problem in Terence Tao's book about analysis and i got stuck trying to prove this by mathematical induction.   2.2.5 -> Prove the following proposition:  (Strong principle of induction) Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m \geq m_0$, we have the following implication : if $P(m')$ is true for all natural numbers $m_0 \leq m' \leq m$, then $P(m)$ is also true. Then we can conclude that $P(m)$ is true for all natural numbers $m \geq m_0$.  He gives the hint to define $Q(n)$  to be the property that $P(m)$ is true for all $m_0 \leq m \leq n$  I begun doing a induction on $n$ with $m_0$ as a base case, but when i got to prove that given that $Q(k)$ implies $P(k)$ then to prove that $ Q(k+\!\!+)$ implies $P(k+\!\!+)$ i got stuck. The $+\!\!+$ is the increment operator that is basicaly a succesor function.
I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of natural numbers, order properties of natural numbers, the trichotomy of order for natural numbers and principle of induction.  Proposition 2.2.14 (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m ≥ m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 ≤ m' &lt; m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m ≥ m_0$. Exercise 2.2.5. Prove Proposition 2.2.14. (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0 ≤ m &lt; n$; note that $Q(n)$ is vacuously true when $n &lt; m_0$.)  Thanks!
In Bash, when to alias, when to script, and when to write a function? It's taken me almost 10 years of Linux usage to ask this question. It was all trial and error and random late-night internet surfing.  But people shouldn't need 10 years for this. If I were just starting out with Linux, I'd want to know: When to alias, when to script, and when to write a function?  Where aliases are concerned, I use aliases for very simple operations that don't take arguments.   alias houston='cd /home/username/.scripts/'   That seems obvious. But some people do this:  alias command="bash bashscriptname"   (and add it to the .bashrc file)   Is there a good reason to do that? I'm trying really hard, but I genuinely can't think of any circumstances in which I'd want to do that. So, if there is an edge case where that would make a difference, please answer below.  Because that's where I would just put something in my PATH and chmod +x it, which is another thing that came after years of Linux trial-and-error.  Which brings me to the next topic. For instance, I added a hidden folder (.scripts/) in the home directory to my PATH by just adding a line to my .bashrc (PATH=$PATH:/home/username/.scripts/), so anything executable in there automagically autocompletes.  If I needed to.  I don't really need that, though, do I? I would only use that for languages that are not the shell, like Python.  If it's the shell, I can just write a function inside the very same .bashrc:  funcname () {   somecommand -someARGS "$@" }   As I stated, I found a lot of this out through trial and error. And I only truly saw the beauty of functions when my computer died and I was forced to use the computers of the people around me when they weren't using them.   Instead of moving a whole directory of scripts from computer to computer, I ended up just replacing everyone else's .bashrc with my own, since they had never even made a single modification.  But did I miss anything?   So, what would you tell a beginning Linux user about when to alias, when to script, and when to write a function?   If it's not obvious, I'm assuming the people who answer this will make use of all three options. If you only use aliases, or only use scripts, or only use functions—or if you only use aliases and scripts or aliases and functions or scripts and functions—this question isn't really aimed at you. Singular to plural noun Many nouns that end in ‑f are made plural by changing the ‑f to ‑v‑ and adding ‑es.  +----------+-----------+ | Singular |   Plural  | +----------+-----------+ |  half    |  halves   | |  leaf    |  leaves   | |  shelf   |  shelves  | +----------+-----------+     But some nouns that end in ‑f are made plural simply by adding ‑s.  +----------+-----------+ | Singular |   Plural  | +----------+-----------+ |  chief   |  chiefs   | |  roof    |  roofs    | |  cliff   |  cliffs   | +----------+-----------+   Some nouns that end in ‑f can be made plural in two ways, either by adding ‑s or ‑ves:   +----------+--------------------+ | Singular |      Plural        | +----------+--------------------+ |  scarf   |  scarfs or scarves | |  hoof    |  hoofs or hooves   | |  dwarf   |  dwarfs or dwarves | |  wharf   |  wharfs or wharves | +----------+--------------------+   Now my question is how to determine whether to use only ‑s or to change the ‑f to ‑v‑ and add ‑es? Prime number expressible as $2^2+3^3+5^5+7^7+11^{11}+\cdots$ I came across this statement while studying electric currents and I am confused:  "There is no electric field inside a conductor. Hence no current can flow through it".  Is there a fallacy in this statement?  Any help would be appreciated.  The conductor being discussed is ideal. What is the driving force for the current in this case? Symmetry condition for continuous coordinate transformations Can I get individual man pages for the bash builtin commands? Is there anywhere you can download a manpage for every builtin commands?  I know you can just use help or man bash and search to find info about it, but I want them separated, so I can just do man read and get the read manpage. I can't seem to find a satisfactory answer to this anywhere on the web. While  poses a related question, I don't think it sufficiently answers mine...  During the opening chapters of Jedi Search, Luke addresses the new Senate on the need to find new Jedi talent, or Force sensitive individuals. During this proposal, he cites that the most fruitful leads of Jedi potential are in the descendants of Jedi Knights: as Force sensitivity is hereditary. The book even discusses the notion that Leia should be comparatively well attuned to the Force, due to her being the child of Anakin Skywalker.  I understand that the concept of the Jedi as a totally celibate and monastic order was something that was retconned into the lore after Episode II: Attack of the Clones, and that there are examples of entire nuclear Jedi families exist (such as Andur Sunrider and Plo Koon)...  But if Force-manipulative ability is passed down genetically (as a potentially mutative property of their genetic makeup), why would the Jedi order be so numerous if they practice Chastity?  Even if procreation is not expressly forbidden (as in the pre-retcon Expanded Universe): why would Jedi need special permission to produce more Jedi potential children?  How are more Jedi created, if not through procreation? Has their profusion been deliberately retarded, and if so why? is there another explanation for their (pre-empirical) plenitude? Can't kill a sleeping process Are there rules about using "that" to join two clauses? He will understand that I was not joking.   He will understand I was not joking.  Which of the sentences is correct? Are there any specific rules about the use of &quot;that&quot; in the sentences I reported as example? When displaying the value of a decimal currently with .ToString(), it's accurate to like 15 decimal places, and since I'm using it to represent dollars and cents, I only want the output to be 2 decimal places.   Do I use a variation of .ToString() for this? How do you remove a default program association for file types in Windows 7? 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