I saw somewhere the the cantor set is in bijection with $\mathcal P(\mathbb N)$, so this mean that $\mathcal P(\mathbb N)$ is uncountable. But how can I show that $\mathcal P(\mathbb N)$ is uncountable ?
Consider a common proof of the uncountability of $(0,1]$, as presented  for example:  We assume that the reals can be arranged in a sequence $x_k$, represent every number in $x_k$ by its nonterminating decimal expansion, use the diagonal argument to create a sequence of decimal digits that differs from the nonterminating decimal expansion of each number in $x_k$ and then claim that this sequence of digits so generated must be the decimal expansion of some number not in $x_k$.  My problem is with this last step. Could it not be the case that the new sequence of digits is just the terminating decimal expansion of some number already in $x_k$?  Would it not be more correct to use the sequence $x_k$ to generate a sequence of all (both terminating and non-terminating) decimal expansion of all numbers in $[0,1)$ and then apply the diagonal argument?
Return to 1st image on slideshow What's the subject of "There is my biscuit!" ? And how about "There is one biscuit left"? What's the subject, grammatically speaking, of these sentences?   There is my biscuit! My biscuit is there! There is one biscuit left.   I don't really know how to analyze these. The following observations seem to be relevant, though:   The verb to be doesn't take an object. In the first two sentences, there is a demonstrative pronoun. It is therefore potentially a subject, I think. I don't know what part of speech is taken by there in the third sentence. In favour of reading my biscuit and one biscuit as the subjects is that if there were more than one biscuit then, in each case, the verb would become plural (There are my biscuits! etc.) In favour of reading one biscuit as the subject of the third sentence is that if we recast it as (the slightly unidiomatic) There remains one biscuit, it seems that one biscuit would have to be the subject.   All of this leads me tentatively towards thinking that my biscuit and one biscuit are the subjects here, but I'm not totally confident about this.  One other possibility: since to be doesn't take an object, does that mean it can have two subjects? In I am he, presumably both I and he are subjects, if neither is an object? $$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$  when f is continuous on $[0,1]$  I know it can be proved using bounded convergence theorem but,  I wanna know proof using only basic properties of riemann integral and fundamental theorem of calculus and MVT for integrals ...   Thank you. apex:inputField can only be used with sObjects I am trying to create a custom visualforce page that saves new opportunity contact roles to an opportunity. I am having trouble creating the first step of listing out all contacts that relate to that account. The error I am getting is...     Save error: Could not resolve the entity from  value binding '{!u.contact}'.   can only be used with SObjects, or objects that are Visualforce field component resolvable.   I do not understand why I am receiving this error. u.contact should resolve to a Contact object, which is an sObject. Am I misunderstanding something?  Here is my code...  Controller  public class ChooseContactController  {     public String accountId = ApexPages.currentPage().getParameters().get('accountId');     public Account account { get; set; }     public List&lt;UnfundedOpportunityContactRole&gt; unfundedOpportunityContactRoles { get; set; }      public ChooseContactController()      {          unfundedOpportunityContactRoles = new List&lt;UnfundedOpportunityContactRole&gt;();         List&lt;Contact&gt; contacts = [SELECT Id, Name FROM Contact WHERE AccountId =: accountId];         account = [SELECT Id, Name FROM Account WHERE Id =: accountId];          for(Contact c: contacts)          {             unfundedOpportunityContactRoles.add(new UnfundedOpportunityContactRole(c, 'The Man'));         }     }      public void Save() {}     public void Create() {}      public class UnfundedOpportunityContactRole     {         public Contact contact { get; set; }         public String role { get; set; }          public UnfundedOpportunityContactRole() { }          public UnfundedOpportunityContactRole(Contact contactParam, String roleParam)         {             this.contact = contactParam;             this.role = roleParam;         }     } }   VF page  &lt;apex:page controller="ChooseContactController"&gt;   &lt;apex:form &gt;     &lt;apex:pageBlock title="Choose a contact" mode="edit"&gt;       &lt;apex:pageBlockButtons &gt;         &lt;apex:commandButton action="{!create}" value="New"/&gt;         &lt;apex:commandButton action="{!save}" value="Save"/&gt;       &lt;/apex:pageBlockButtons&gt;         &lt;apex:pageBlockSection title="{!account.Name}"&gt;           &lt;apex:dataTable value="{!unfundedOpportunityContactRoles}" var="u"&gt;             &lt;apex:column headerValue="Contact"&gt;               &lt;apex:inputField value="{!u.contact}" /&gt;             &lt;/apex:column&gt;             &lt;apex:column headerValue="Role"&gt;               &lt;apex:inputField value="{!u.role}" /&gt;             &lt;/apex:column&gt;           &lt;/apex:dataTable&gt;         &lt;/apex:pageBlockSection&gt;     &lt;/apex:pageBlock&gt;   &lt;/apex:form&gt; &lt;/apex:page&gt; When is the present perfect tense used instead of the past tense? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. use of commas with conjunction reduction I wimped out in answering , dropping the commas and going to parentheses in this sentence, because I was not sure of the placement/correctness of commas in this construction. Now I'd like to get clear on how to use commas in this sentence (without reformulating it...no moving "poems", for example), and why. I'm not even sure that conjunction reduction is what causes the awkwardness here...  Wimp version:     I like Lord Byron's (and also enjoy a number of Percy Shelley's) poems.   Now I'm asking, if you use the comma, do you use both commas? It looks mighty strange with "poems" out there by itself:     I like Lord Byron's, and also enjoy a number of Percy Shelley's, poems.   ...Or is the comma even required? I don't like this, but is it correct?     I like Lord Byron's and also enjoy a number of Percy Shelley's poems. A little brain fart What is a nicer, less immature saying?  I love using 'A Freudian slip', but that is only applicable in certain situations. How to show $a,b$ coprime to $n\Rightarrow ab$ coprime to $n$? Using exec and tee to redirect logs to stdout and a log file in the same time In a bash script, how can I redirect all standard outputs to a log file and tee the output on the screen using exec ?  log_file="$HOME/logs/install.txt-`date +'%Y-%m-%d_%H-%M-%S'`" [ -f "$log_file" ] || touch "$log_file" exec 1&gt;&gt; $log_file 2&gt;&amp;1   This code redirect all the log to the log file but not to the screen . I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help How to call a constructor on an already allocated memory? How can I call a constructor on a memory region that is already allocated? Summing data from column in the Attribute table I'm working with population data, and I have an issue when it comes to knowing the sum of a population inside the attribute table.  In my attribute table, I'd like to add all the numbers in the red box, i.e. all the numbers together from the column &quot;UN_2015_E&quot;. Is there a way to do so? Simple property of a valuation on a field Asked again to review a paper, when the authors don't wish to modify it I have received a manuscript to review for a journal. The interesting thing is, I had already reviewed this article (exact same title, abstract and author list, almost exact same content) for another journal a few weeks ago where I recommended that it be published, but only after major revisions of both form and content. When I first reviewed it, I wrote a two-page review, listing some questions and several “actionable” comments, ranging from some concerns about exactness of the text (some conclusions didn't seem fully backed by the results) all the way down to trivial stuff (grammar, a few typos, graphic issues with the figures, etc.).  Now, the manuscript has come to me for review again, but it is almost unchanged from the first version. None of the serious stuff has been addressed, and even most of the trivial stuff was not fixed (there's at least one remaining typo, and the figures still aren't fully legible). However, I think this behavior from the authors is clearly a bad signal, which should be somehow conveyed to the editor: they're not willing to amend their work, and would rather do some journal-shopping.  What would you suggest me to do? Should I just re-send my earlier review? Add a note to the editor about my knowledge of the “history” of the paper? Or maybe even include it in my review, so the authors are aware that people know of their behavior, and maybe feel bad enough to change their ways?    PS: I wrote it in the present tense, but it's actually a story from my past. I'm not sure I did the right thing at the time, and I think it's better to actually formulate it as an open question… I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this:  $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that I need to find are just 3: $a, b, c$; however the points I have are $\#N&gt;3$ and so it looks like the determination of the coefficients is impossible because is over-determined. Does anyone have an idea of what should be done in such case (supposing it is even possible)? When reviewing Suggested Edits on Stack Overflow, I often come across "fixed formatting" suggestions that use  to place emphasis on certain keywords, but isn't actual code.  For example:     I am having a difficult time with a background task in iOS. The problem I seem to be facing is that iOS is silently terminating the App if my background task runs for too long. What can I do to increase time iOS will wait for my background task to complete?   Gets edited to:     I am having a difficult time with a background task in iOS. The problem I seem to be facing is that iOS is silently terminating the App if my background task runs for too long. What can I do to increase time iOS will wait for my background task to complete?   This seems like an invalid use of an inline code span because, well, it isn't code, and it is distracting. There are a    of these edits being made. Often enough they get approved, which reinforces the behavior.  Is this a valid use of inline code spans? If not, should it be edited and corrected? Should edits suggesting these changes be rejected? wpa_supplicant store password as hash (WPA-EAP with phase2="auth=PAP") Is there a way to store my password in /etc/wpa_supplicant/wpa_supplicant.conf as some hash instead of plaintext?  By "password" I refer here to the password used for phase2 authentification. I do not refer to the Preshared Key (PSK) which could be hashed using wpa_passphrase.  For phase2 MSCHAPv2 or MSCHAP authentification I could store the password as MD4 hash using nt_password_hash (see example wpa_supplicant.conf line 659).  Is there any way to store my PAP password as hash?  Or: Is there a way to store the password in some sort of external storage? The example wpa_supplicant.conf suggests the use of such an external storage (using ext:????) but I could not find any documentation about it.  I am aware that storing the password as hash does not increase wifi security. But as the password MUST be the same as the password for other services (account management, subscriptions, payments ...) I don't want it to be stored as plaintext. How to install KDE along Unity? I have just installed the latest Ubuntu 14.04 and I would like to install KDE along side my Gnome/Unity that I have now. How can I do this?  Thanks! If you have a glass of water, say, three quarters full and you pour it at an angle of say, $45^{\circ}$ with respect to the the table, the water comes out of the glass and goes directly down towards the floor.  However, when the glass is more full, or even three quarters full and the 'angle of pouring' is far less with respect the table, when the water comes out of the glass, rather than going straight down it kind of stays stuck to the glass and travels down the outside of it.  Why does this happen? (I'm sure many of you have observed it yourself by accident and ended up making a mess). Is There a Difference Between Taxonomies and Categories? I've been reading the codex for WordPress closely and am confused about the vague definitions between what a Taxonomy is and what a Category is. For that matter, term is also a word that is used in the description of Taxonomies, but seems used in a manner that hints that it isn't solely limited to a Taxonomy.   Both are described as a way to classify and group together posts. I do understand that Taxonomies are a more general form of categories, but how exactly do they relate to each other?   Is there a subtle difference I'm just not picking up on?