I was watching a  on Banach-Tarski, which has a preamble section about Cantor's diagonalization argument and Hilbert's Hotel. My question is about this preamble material. At c. 04:30 ff., the author presents Cantor's argument as follows. Consider numbering off the natural numbers with real numbers in $\left(0,1\right)$, e.g. $$ \begin{array}{c|lcr} n \\ \hline 1 &amp; 0.\color{red}50321642239817 \ldots \\ 2 &amp; 0.0\color{red}7829136011205 \ldots \\ 3 &amp; 0.31\color{red}11370055629 \ldots \\ 4 &amp; 0.999\color{red}9261457682 \ldots \\ 5 &amp; 0.0001\color{red}042507334 \ldots \\ \vdots &amp; \vdots \end{array} $$ Then you could form a new real in $\left(0,1\right)$ not already in the list, e.g. $0.\color{red}{68281} \ldots$. Hence there are more reals than naturals. I have two questions about this:  Couldn't you run exactly the same argument (erroneously) for rational numbers in $\left(0,1\right)$? E.g. say I choose powers of $\frac{1}{2}$, giving:  $$ \begin{array}{c|lcr} n \\ \hline 1 &amp; 0.\color{red}4999999999999 \ldots \\ 2 &amp; 0.2\color{red}499999999999 \ldots \\ 3 &amp; 0.12\color{red}49999999999 \ldots \\ 4 &amp; 0.062\color{red}4999999999 \ldots \\ 5 &amp; 0.0312\color{red}499999999 \ldots \\ \vdots &amp; \vdots \end{array} $$ So $0.\color{red}{55555} \ldots$ is not in the list, suggesting that the cardinality of the rationals is greater than that of the naturals. But a different argument shows that their cardinalities are the same. So there seems to be something wrong with the diagonal argument itself?  As a separate objection, going back to the original example, couldn't the new, diagonalized entry, $0.68281 \ldots$, be treated as a new &quot;guest&quot; in Hilbert's Hotel, as the author later puts it (c. 06:50 ff.), and all entries in column 2 moved down one row, creating room?  Admittedly, you could diagonalize this expanded list again; but then you could also move the guests down again. So the argument does not seem to show that there's any fundamental problem, i.e. that you can't continue pairing off the reals with the naturals forever?
If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not apply? Is it because we can't be certain that the number produced is a rational number?
I am wondering whether there exists a function such that:  $$\lim_{x \rightarrow a}f(x)=\infty$$  at some point $a$ on the real axis but yet,  $$\int_{-\infty}^{+\infty}\left|f(x)\right|\ dx&lt;\infty$$  Does the fact that a function is unbounded imply that it has no finite integral? example:   Does Chocolatey run the native program installer? If I prefer simplicity, should I avoid packages with (Install) in their name? Using v2.73, cycles and a very simple scene: an array of cubes, no big deal.   If I set the shader's material to transparent, the cubes in the foreground do become transparent but those far away in the back do not disappear completely and this strange dark things appear (visible in rendered mode and in final renders)    I've played with the clipping distance on the viewport and camera, but I can't figure out how to make those ghosts go away... How does IPv4 Subnetting Work? How do I change how executable files are handled by the file manager? When I double click on text files, nautilus asks me what I would like to do with them. I find this to be intrusive, as I just want them to open.  How can I fix this?  Note: This question is a "canonical" question on the subject; edit if needed. Note 2: Text files can be HTML, CSS, TXT, and more. Prove that $d(n)\leq 2\sqrt{n}$ In Waclaw Sierpinski's book Elementary Theory of Numbers on page 168 there is the following exercise:  "Exercises. 1. Prove that for natural numbers $n$ we have $d(n) \leq 2\sqrt{n}$," where $d(n)$ is the number of divisors of n.  As a hint right below is given: "The proof follows from the fact that of two complementary divisors of a natural number $n$ one is always not greater than $\sqrt{n}$.  I understand the hint but I don't know how it can be used to prove $d(n)\leq 2\sqrt{n}$.  Complementary divisors are pairs of divisors that when multiplied gives the number that is to be divided. For example the number $120$ has the complementary divisors: \begin{align*} &amp; 1, 120 \\ &amp; 2, 60 \\ &amp; 3, 40 \\ &amp; 4, 30 \\ &amp; 5, 24 \\ &amp; 6, 20 \\ &amp; 8, 15 \\ &amp; 10, 12 \\ \end{align*} How do you prove that $d(n) \leq 2\sqrt{n}$? Case-sensitive Google search Is there any way to make a Google search case-sensitive? I'm using Ubuntu Maverick and Virtual Box 3+. I have Windows 7 Ultimate. The problem is that it doesn't detect USB drives in Windows 7, but USB peripherals (mouse + keybord) are working.  I followed , but I can't find the user group vboxusers. How can I continue? I would also like to upgrade to the latest version without any risk (without losing my virtual hard disk drive). Difference between "right of=" and "right=of" in PGF/TikZ In the PGF/TikZ manual, sometimes I see the option right of=somenode instead of right=of somenode. They look very similar, but the effects are different.  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And can I also get a list of these duplicated posts? How to find P(2X>4)? This is a Binomial question. I am required to find P(2X>4), and given that X~B(10,0.2), how do I convert it to 2X~B(n,p)?  Have tried converting from P(2X>4) to P(X>2), but understand that it doesn't work that way.  Do i have to use integration or any formula that i should refer? A secretary with ____ good knowledge of English: "a" or no article? I'm trying to pass a test on "Articles exercise" as they call it . Reviewing my answers I got stuck about the two I posted on the picture below.  As I understand both words are uncountable (according to the ), so it looks like (or said for example ) I can't say a/an with an uncountable noun.  But this test says the opposite. Is there some trick? Or am I just dumb? Please help me to understand why I need to use "a" article in these cases?        We need a secretary with ____ good knowledge of English.          a  (marked as the correct answer)   the     a / the     no article is needed (the answer I think is correct)        We're having ____ terrible weather.         a (marked as the correct answer)     the     a / the     no article is needed (the answer I think is correct) Greets  This is exercise 15.d chapter 3 of Stein &amp; Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, instead I knew how to do it with the Casorati-Weiestrass Theorem, here is my answer:  Define $g(z)=f(1/z)$ for $z\neq{0}$,then by the hypothesis we must have that for any $\epsilon&gt;0$ $g(D_{\epsilon&gt;0}(0)-\{0\})$ is not dense in $\mathbb{C}$, then the singularity at $0$ of $g$ is not essential, this implies $f$ must be a polynomial, but if $f$ is a non-constant polynomial, it is easy to see that its real part must be unbounded, so $f$ must be constant.  I would like to know an answer with the the maximum modulus principle.  Thanks