Here is my rather simple idea. I will treat the set of real numbers as a set of discrete continuities, each separated by an Epsilon ball that tends to 0.  So, let's say P(b) is true. We then assume P(k) is true, and prove that P(k+e) is true, where e goes to zero.  I just want to know if this is a valid technique or not because our teacher said that mathematical induction can only be applied to discrete structures, but I see no difficulty in treating a continuous system as a set of infinitesimal discrete quantities. Mak
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.  Of course you have to change some things in the inductive step, when you want to use it on real numbers.  I guess that using induction on real numbers isn't really possible, since $[r,r+\epsilon]$ with $\epsilon > 0$, $r \in \mathbb R$ is never empty.  Can you either give a good reason, why it isn't possible or provide an example where it is used?
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) Taxonomy count per Post type Below is the code i use to output the post tag (taxonomy) count/number. I want to be able to split the count based on post type that the taxonomy features in (rather than the total number).   So i have the default "post" Post type, aswell as "blogs", &amp; "pics". I want the taxonomy count to display something like: x posts | x blogs | x Pics              &lt;?php             $tags = get_tags( array('name__like' =&gt; "a", 'order' =&gt; 'ASC') );             foreach ( (array) $tags as $tag ) { ?&gt;                 &lt;li&gt;                                                 &lt;a href="&lt;?php echo get_tag_link( $tag-&gt;term_id ) ?&gt;"&gt;                                   &lt;span class="name"&gt;&lt;?php echo $tag-&gt;name ?&gt;&lt;/span&gt;                         &lt;span class="number"&gt;&lt;?php echo $tag-&gt;count ?&gt;&lt;/span&gt;                     &lt;/a&gt;                 &lt;/li&gt;             &lt;?php } ?&gt; Books on Number Theory for Layman Omitting the hypotheses of finiteness of the measure in Egorov theorem I want to prove that if I omit the fact that $\mu (X) &lt; \infty$ in Egorov theorem and place instead that our functions $|f_n| &lt;g$ and $g$ is integrable, we still get the result of Egorov's theorem.  Fix $m$ a natural number.  I took $ w_{n} = |f_n-f|$ and thus by DCT $\int |f_{n} - f|$ goes to zero.  Then I took $\bigcup_n {( w_{n} \geq 1/m)}$. I need its measure to be finite. Its measure is less than the sum of the measures of each $ w_n\geq 1/m)$ varying $n$, and by Tchebychev, this is less than $m\int|f_{n} - f|$.  But I got stuck here.  Any help is appreciated. Thanks! Existence of minimal prime ideal contained in given prime ideal and containing a given subset Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$.   My approach: Let $\langle X\rangle$ be the ideal generated by $X$. $P^c$ is a multiplicative set, as $P$ is prime. $\langle X\rangle\subseteq P\subseteq R$ and $P^c\cap\langle X\rangle=\emptyset $ so by Zorn's lemma there exist prime ideal $I_1$ satisfying $\langle X\rangle\subseteq I_1\subseteq P$. As $I_1$ is prime, it's complement is a multiplicative subset which does not intersect $\langle X\rangle$, hence I can find prime ideal $I_2$ satisfying $\langle X\rangle\subseteq I_2 \subseteq I_1$ and so on... I've got an infinite descending chain of prime ideals and I can't be sure that the process will stop.  Hints will be appreciated, thank you.   Edit: Can I conclude that every prime ideal contains a minimal prime ideal? I have configured my MySQL Database to require passwords on all users, even root from the machine itself.  Now I discovered that there are empty Users in my Database  Reproduce: mysql -u root -p and then use mysql; &amp; select * from user;  It gives me 2 entries, one with "localhost" and user &lt;empty&gt; and one with the machine's hostname and &lt;empty&gt;.  Now I tried to access the account with mysql -u ' ' (yes it's correct, leave a space between the ' things) and I log in without a password. The user can "only" see information_schema and test, the two databases created by default. He does not has access to mysql or any other custom created databases.  I already changed the Password of this user to something I won't tell you  mysql -u root -p use mysql; UPDATE mysql.user SET Password=PASSWORD('thisisasecretpassword') WHERE USER='root'; FLUSH PRIVILEGES;   Now my Questions:   Is any MySQL Server vulnerable for an attack with this entry? Could an attacker break out of this account or the two default databases? Should I password-protect those Users or can I delete them? Are They required for some MySQL-internal things? "Hail Mary", "Glory Be" in the plural Possible Duplicate:          In the rosary, a Catholic sacramental composed of prayer beads, there is a short prayer concluding each decade called "Glory Be" ("Glory be to the Father, Son and Holy Spirit...").  Would the plural be Glory Bes or Glory Be's?  Likewise, do you say 10 Hail Marys? How do I use a different drive as an installation location for Steam? I recently got an invitation for Steam. I want to download games but my SSD drive which I have Ubuntu will not be enough. How can I install Steam on a different hard drive?  I went into the "Steam Library Folders" but I can't delete the original folder so steam can install on that drive. I have also tried to change the place but it is again on the down side: I saw this movie a while back. I don't remember how it starts, but from where I remember watching it, it may have not been too far from the start. Apparently, Earth has been conquered by some alien empire or something and humanity are basically cavemen being used as slaves. From what I remember, the main antagonist alien is seen taking one of the slaves to a room where a machine is used on him to make him smart; this is apparently a big no-no so the main antagonist and his sidekick have to keep it quiet. The idea was that by making one of the slaves smart, it could increase the work output of the others. Apparently the antagonist is using the slaves on Earth to gather gold bullion  which is apparently still very valuable to the aliens but he's doing it illegally (not the slave labor; that's apparently ok). The protagonist who had the machine used on him becomes the intermediary between the slaves and the aliens and begins teaching the other slaves. I remember him teaching them mathematics and I remember him saying that it was universally accepted (I think he may have been referring to algebra or trigonometry). At some point he becomes aware that they are slaves and organizes a plan to free not only his fellow slaves but the Earth. Some escape and find an old museum which has old flight simulators, and shows that humanity wasn't always a slave race and that it may have been modern times in which Earth was invaded (i.e. no lasers or starships). I think they also find Fort Knox as there is a huge collection of gold bricks and weapons. The protagonist comes up with a plan to destroy some sort of teleporter the aliens use to transport stuff to and from Earth, however that alone wont be enough as they could fight back, which will be bad for humanity, which is still undeveloped. So the plan also includes using the teleporter before destroying it to to transport and detonate a nuclear warhead on the alien home world which apparently cripples any attempt to immediately send a fleet but also as a show of strength. After executing their plan, the main antagonist is captured by the humans and as an insult is locked in a cage surrounded by the gold he sought. If I recall correctly the place is set to detonate as to destroy the precocious gold and that will be used against the aliens should they mobilize a fleet in the future (i.e. attack us and we destroy Earth's supply of gold). The antagonist's side kick for some reason I remember as a big black man from whom the antagonist had removed his hand at some point and also apparently the antagonist was cruel to him. After the antagonist is locked up, he is trying to appeal to his sidekick's loyalty, however the sidekick takes the humans' side, I'm also getting an image that apart from treating him kindly the sidekick's missing hand was restored. Also schemas can communicate over database links to schemas in other remote databases.    Could one replace the preposition over with through? Will it change the meaning, and if so, when should we use over instead of through? Berry naming scheme list of LaTeX font families Why isn't $\mathbb Z [\sqrt{pq}]$ a factorial domain This is exercise 8 on page 147 of Jacobson's Basic Algebra:     Let $p$ be a prime of the form $4n+1$ and let $q$ be a prime such that $ \left( \frac{q}{p} \right) =-1$. Show that $\mathbb Z[\sqrt{pq}]$ is not factorial.   Here, $ \left( \frac{q}{p} \right) =-1$ means that $\overline{q}$ is not a quadratic number in the field $\mathbb Z / (p)$.  In the special case when $p=5$ and $q=2$, $9 = 3 \cdot 3 = (\sqrt{10} +1 )(\sqrt{10 }-1)$, with $3$ and $\sqrt{10} +1$ or $\sqrt{10} - 1$ nonassociates. But I have no idea what can I do with the general case.  I know that primes of the form $4n+1$ can be written in the form $m^2 + n^2$ with $m,n \in \mathbb Z$. But when $q$ is odd and not of the form $4n+1$, $pq$ is not the sum of two quadratic numbers. How can I use the fact $\left( \frac{q}{p} \right) =-1$?  Thanks~ How secure would github hosting be for private repositories? Contribute to bounty? What is a word/idiom for 'unable to decide'? Let's say I have an important decision to make and I can't decide between two competing things (like break up with a girl or not break up with her). What would be a word/idiom to express that? Can the Universe be deterministic and still allow free will? Possible Duplicate:             As asked in the title, can a Universe which is deterministic at its fundamental level still allow free will? How to use oEmbeds on Post Content during AJAX requests I have an AJAX request that returns a post. The post_content has links to Twitter, YouTube, TED and other platforms that are registered as oEmbed providers in a default, vanilla WordPress install. By now, the WordPress oEmbed handler does not register them and displays plain links and nothing else.   How can I fetch content via AJAX with oEmbed support? Figure reference \ref{fig:result} renders as ?? in PDF How does vendor restocking work in Fallout 4? This is a comic book I read no later than 1993 (so is pre-1993).  And probably after 1988 though I can't be certain.  The protagonist is a huge barbarian with an axe, who seems like an old man, who is defending a series of gigantic walls against an invading army, apparently single-handedly. They keep having to fall back to inner walls. He has to get a massage at one point (not that kind of comic btw).  It's not a super-hero or funny comic, it's more of a serious graphic novel.  That's about it, sorry it's so vague.