What is the difference between a natural join and an inner join?
I am wondering how to differentiate all these different joins ...
My question relates to Chapter 3, Exercise 8 in &quot;Baby Rudin&quot;. It states: If $\sum_n a_n$ converges, and if $\{b_n\}$ is monotonic and bounded, prove that $\sum_n a_n b_n$ converges. My attempt would have been: Since $\{b_n\}$ is monotonic and bounded, $\{b_n\}$ converges and it exists $\inf \{b_n\}$ as well as $\sup \{b_n\}$. But then we have $$ \left| \sum_n a_n \inf \{b_n\} \right| \leq \left| \sum_n a_n b_n \right| \leq  \left| \sum_n a_n \sup \{b_n\} \right| \leq \max \left( \left| \sup \{b_n\} \right|, \left| \inf \{b_n \} \right| \right)\varepsilon \leq \tilde{\varepsilon} $$ since $\sum_n a_n$ converges and $\max \left( \left| \sup \{b_n\} \right|, \left| \inf \{b_n \} \right| \right)$ is a finite number. This would imply, by the comparison test, that $\sum_n a_n b_n$ converges as well. $\quad \Box$ But as there was a way longer, more rigorous proof chosen in this , I'm a bit suspicious that my proof is not complete. Am I missing something?  EDIT: Thanks everyone! @hermes: The last part of your proof gave me the following idea. As $\lim_{n \to \infty} b_n = c$ and $\{b_n\}$ is monotonic, couldn't we just set $b_n = c - c_n$ with a monotonically decreasing sequence $\{c_n\}$ which has $\lim_{n \to \infty} c_n = 0$. Then we have $$\sum_n a_n b_n = \underbrace{c \sum_n a_n}_{\text{converges by assumption}} - \underbrace{\sum_n a_n c_n}_{\text{converges by Theorem 3.42}} \leq \varepsilon_1 - \varepsilon_2 = \varepsilon $$ since  Theorem 3.42 Suppose  the partial sums of $\sum_n a_n$ form a bounded space $\quad \checkmark$ $c_0 \geq c_1 \geq c_2 \geq \dots \quad \checkmark$ $\lim_{n \to \infty} c_n = 0 \quad \checkmark$  Then $\sum_n a_n c_n$ converges.  Thus $\sum_n a_n b_n$ converges as well. $\quad \Box$ Now that should hold, I think. So I wouldn't need to go through all the estimates. Scala Lists of lists of integers I'm currently making a Minesweeper in C#. I've already done the basics (generating minefield, rules and ui) but I don't know how I should implement when the user clicks a cell and the cell has no mines nearby and clicks nearby cells with also no mines nearby, like this:    And after click    This is my current code on this:  if (MineField(x,y) == false) // If the clicked cell isn't a mine {     if (GetNeighbours(x,y) == 0) // GetNeighbours returns the amount of mines in the closest 8 cells     {         // To be honest, no idea...     } } The left-nav is great - I like it, users who don't can disable it. Perfect world. Now it's stickied it's always there too, which is really useful.  When I'm navigating in areas not directly linked to in the left nav and have scrolled down, I often want to re-use the other navigation. This means scrolling up. Which is fine, of course it's how websites work.  However, if the left-nav had a "Top" link which took the user to the very top of the page it would be great for getting back to specific contextual menus and things that are not linked to in the left-nav.  For example, if I'm viewing questions on a specific tag and I've scrolled down, I sometimes want to get back up to the top to remove a tag, or change from "Latest" to "Featured" etc. Although clicking the "Questions" link in the left-nav takes me back to the top, it removes the tag(s) I was searching with.  There are other use cases where this would be useful, like long questions and answers (Etc). So I think a "Top" link in the left nav would be very useful.    I think this is different to the  as it doesn't mention the left-nav, and my question is suggesting this idea because we have a good place to put it and it'll be a useful addition to the left-nav. Visualization for Euclidean Algorithm How can I find non-ASCII characters in MySQL? I'm working with a MySQL database that has some data imported from . The data contains non- characters (em dashes, etc.) as well as hidden carriage returns or line feeds. Is there a way to find these records using MySQL? Idea behind proof that $\frac{21n+4}{14n+3}$ is irreducible for all $n$ Are the decimal places in a CSS width respected? What is the command to run System Settings from a terminal? System Settings can be run from the launcher (pinned by default), the Dash, or the power cog. But what command would I enter in a terminal window if I want to run it from there? has the following ability:     Whenever a creature you control with a +1/+1 counter on it dies, return that card to the battlefield under your control at the beginning of the next end step.   I had a number of creatures on the battlefield, including Marchesa herself, with one or more +1/+1 counters on them when  was played, destroying each creature on the board.  When Marchesa and several other creatures would be killed simultaneously,   Does Marchesa's ability trigger for herself? Does Marchesa's ability trigger for the other creatures under my control with +1/+1 counters?   In attempting to resolve the issue, we tried to look things up, leading to , which cites Rule 603.6d as proving that Marchesa's ability should trigger both for herself and for each other creature with +1/+1 counters.  Unfortunately, none of us could actually understand Rule 603.6d, aside from this example:     Two creatures are on the battlefield along with an artifact that has the ability Whenever a creature dies, you gain 1 life. Someone plays a spell that destroys all artifacts, creatures, and enchantments. The artifacts ability triggers twice, even though the artifact goes to its owners graveyard at the same time as the creatures.   The example certainly seems analogous to me, but without understanding the actual rule, rather than just the example, we weren't too sure.  A complete answer has to not only answer the question, but back up that answer with citation and explanation of the official rules on the matter. Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$ Why Listen to Music, why not Listening Music Can I make any injective function bijective by patching the codomain? How should I refer to a friend who is a girl but not a girlfriend? When I'm talking about my friend, who is a girl, but not a girlfriend, what word or phrase should I use? If the gender was unimportant, it would not be a problem. But if I want to note that the friend is female, not male, how should I say that, to avoid ambiguity? How to mount an internal HDD on start-up? I have a 500GB internal HDD where I store all my music and photos but it is annoying to manually mount it after a restart and I was wondering how to mount it automatically and make it available to Rhythimbox for playback.  Thanks! How did Harry remove the Philosopher's Stone from the Mirror of Erised? ‘How did I get the Stone out of the Mirror?’   ‘Ah, now, I’m glad you asked me that. It was one of my more brilliant ideas, and between you and me, that’s saying something. You see, only one who wanted to find the Stone – find it, but not use it – would be able to get it, otherwise they’d just see themselves making gold or drinking Elixir of Life.’   Philosopher's Stone - page 217 - Bloomsbury - chapter seventeen, The Man With Two Faces - Harry talking with Dumbledore   Dumbledore doesn't directly answer Harry's question, but rather describes under what circumstances a person could collect the Philosopher's Stone from the mirror. Did Harry have to do anything to retrieve the stone, such as reach into the mirror and take it? If not, what kind of magic could have removed the stone from the mirror and put it into Harry's possession?  A canon-based answer is preferred, such as the books, interviews with J.K. Rowling, or info from Pottermore, but subjective answers within the spirit of canon are welcome. We all know how to blow up the first Death Star: Fire a couple of proton torpedoes into the thermal exhaust port.  It's ray shielded, so only torpedoes will do.  To do this, the Rebel pilots had to fly down a very long trench, where they were vulnerable to fire from any TIE fighters to the rear.  In fact they were so vulnerable that most of the pilots on this run were blown up once the Imperial forces started responding to the attack. This battle is in space, so the fighters can, if necessary, come to a dead stop and hold their position while firing.  Also, when Luke fired his torpedoes, they took a 90 degree turn and went straight into the port.  So the torpedoes didn't have to be fired from in the trench. We also didn't see any gun emplacements firing at the fighters from near the exhaust port, but if there were, the Rebels could have targeted them and taken them out or run interference to make it hard for the towers to fire at a fighter targeting the exhaust port.  (They were already dodging fire in the trench as well.) A fighter could have easily flown directly at the exhaust port, firing torpedoes multiple times on approach and, if not under fire (or if protected by other fighters), could have actually stopped and taken time to target the exhaust port before firing. Is there any valid in-universe reason that the pilots had to fly down the trench before firing on the port?  And if there is, is there any reason they had to enter the trench so early? Just for reference, here is the  on YouTube.  There's no explanation given for having to fly in the trench. setting upper and lower limits in rnorm Is there a program to mount all of my drives automatically? Keyboard macro to change default audio device Windows 7, Logitech G110 keyboard  2 audio devices (headphones and speakers)  I am hoping to find a quick way to macro a key to switch to headphones, and a key to switch to speakers.  Anyone have a good program or anything of that nature?  EDIT:  Using AHK to try and get this task working as hoped is sadly not doing the trick.  I have included a screenshot of my audio set, as well as a screenshot of my script as it sits now.    I have also tried to uncheck "show disabled devices" and modified the script to be Down 1, and Up 1 respectively.  It will work for one device, but it refuses to switch to the secondary.