Let $A$ be a subset of $\Bbb{R}^2$ with the property that every continuous function $f:A\to\Bbb{R}$ has a maximum in $A$. I have to prove that $A$ is compact.  Is $A$ were a subset of $\Bbb{R}$ then we could take a function that is monotone over the components of $A$ and show that each of the components is closed and bounded hence proving $A$ to be compact. But couldn't generalize this idea to a subset of $\Bbb{R}^2$ that easily.
I need to prove that $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function  in $K$ is bounded.  One direction is obvious because of Weierstrass theorem. How can i prove the other direction? I tried to assume the opposite but it didn't work for me.  Thanks a lot.
What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$?  I know that $\nabla f = \left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$ or equivalently $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$.  Eventually, I am going to evaluate this gradient at a point and then dot it with some given unit vector to find a directional derivative. I think maybe what is troubling me is the radical in $f$. Will I have to use the chain rule in order to find the three partial derivatives?  I tried this:  $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{1}{2}}2x$$ $$= \frac{x}{\sqrt{x^2 + y^2 + z^2}}$$  That result looks ugly to me. However, if I did that partial derivative correctly, then I think I know what I am doing. Real Analysis, Existence of a Limit Boundedly:$|\int_0^b\frac{sin(ax)}{x}dx | \le c$ I need to show that there exists a $c$ such that $|\int_0^b\frac{sin(ax)}{x}dx | \le c$, where $a$ is any real number, and $b$ is any real number greater than 0. Also I need to show that the smallest $c$ possible is $c=\int_0^\pi\frac{sinx}{x}dx$. I know that $\int_0^\infty \frac{sin(ax)}{x}dx = sig(a)\frac{\pi}{2}$ where $sig(a)$ literally just means positive one or negative one, depending on the sign of $a$. I think the fourier tranform of the characteristic function on the interval $(0,a)$ is $\frac{sin(ax)}{x}$. After that, I am unsure about how to proceed. Thanks for your help! What is the effect of the number of cross-type focus points on sharp focus? Gnome Panel becomes fully transparent when window is not maximised! Abort Ajax requests using jQuery Is it possible that using jQuery, I cancel/abort an Ajax request that I have not yet received the response from? How to use GROUP BY to concatenate strings in SQL Server? How do I get:  id       Name       Value 1          A          4 1          B          8 2          C          9   to  id          Column 1          A:4, B:8 2          C:9 Abnormal behavior with iwlwifi driver: the device is shown as not working Python: reload component Y imported with 'from X import Y'? All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices.  I started solving problem in this way:    1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a &amp;b\\c &amp;d\end{array}\right]$$ 2) Matrix which commute with matrix $A$ is matrix $B$: $$B = \left[\begin{array}{cc}e&amp; f\\ g &amp;h\end{array}\right]$$ 3) When i solve the equation $AB=BA$, I get this: $$\left[\begin{array}{cc}ae+bg&amp;af+bh\\ce+dg&amp;cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&amp;eb+df\\ga+ch&amp;gb+dh\end{array}\right]$$  How I can get the general look of wanted matrix? What's the easiest way to install a missing Perl module? I get this error:  Can't locate Foo.pm in @INC  Is there an easier way to install it than downloading, untarring, making, etc? Books about matroids Could you recommend any approachable books/papers/texts about matroids (maybe a chapter from somewhere)? The ideal reference would contain multiple examples, present some intuitions and keep formalism to a necessary minimum.  I would appreciate any hints or appropriate sources. Why do mirrorless cameras have quicker autofocus than SLRs using live view? How to know the usage of undocumented function like LinearAlgebra`BLAS? I have a product key for windows 8.1 (Single language edition or not that i can't remember). I have currently windows 10 pro installed.  I am planning to install windows 10 Home so i activate with windows 8.1 key.  So my question is will i be able to activate it if the Product Key is Windows 8.1 Single Language and i install Windows 10 Home(Multiple language edition) or vice versa. Some vertices don't move along with the rig Why do people buy stocks that pay no dividend? Before recently, Apple Inc. did not give dividend to shareholders.  This makes me curious why people buy Apple Inc. stocks; while it might be due to the expectation of selling stocks at a higher price, this can only be possible if other people are thinking the same, thereby increasing demand. But the stocks themselves do not pay interests - so Apple's amazing revenue/profit records would not matter.... Fundamental group of the torus I'm trying to get my head round the following calculation of the fundamental group of the torus, using Seifert Van-Kampen (I know it's easier to do this by considering covering spaces, but I'm trying to learn the Seifert Van-Kampen method).   Consider the torus $T$ as the unit square in $\mathbb R^2$ with opposite edges identified. Let $A$ be an open disc (say of radius $1/4$ about the origin), and $\overline{A}$ be it's closure. Let $B$ be complement of $A$, so that $A \cap B = S^1$. Then we have $T = A \cup B$ where $A$ and $B$ are a neighbourhood deformation retract pair, so we can use Van Kampen. It is clear that $\pi_1(A) = \{e \}$ and $\pi_1(A \cap B) = \mathbb Z$. So we need to find $\pi_1(B)$. Observe that $B$ deformation retracts onto the boundary of the square which, after identification, is homeomorphic to $S^1 \vee S^1$; in this way we see that $B$ and $S^1 \vee S^1$ are homotopy equivalent (inclusion in one direction, the retraction in another), and so $\pi(S^1 \vee S^1) = F_2$ (which can be calculated by Van Kampen, or by using the theory of covering spaces). So, by Van Kampen, $\pi_1(T) \cong F_2 *_{\mathbb Z} \{e\}$. This is the quotient of $F_2$ by the normal closure of the image of $l_* : \pi_1(A \cap B) \to \pi_1(B)$ (where $l_*$ is the homomorphism induced from the inclusion map $l : A \cap B \hookrightarrow B$). The following is causing me confusion: The generator of $\pi_1(A \cap B)$ is homotopic in $B$ to a loop of the form $u.v.\overline{u}.\overline{v}$, where $u$ and $v$ represent the two generators of $F_2$. So $\pi_1(T) \cong \langle a,b ; aba^{-1}b^{-1} \rangle$, the free abelian group on two generators.  Any explanation or geometric intuition would be really appreciated. Thanks Is it possible to play an animation in the viewport at a slow framerate?  Just because of the rather heavy scene it's already going at about 8 fps, but I would like to be able to play it even slower (an in a controlled manner) for the purpose of recording an animation with automatic keyframes. How to get a drive formatted with exfat working? In Ubuntu 13.04 there was an option to use the PPA by Relan in order to get ExFat functionality. Seeing that in Ubuntu this functionality is now available from the repositories without the PPA I tried this but failed to make it work.  So my question is how do I enable exfat support? Remap Keys in Linux I'd like to switch my Ctrl and Alt keys under Linux (to make it more Mac-like).  How do I do this?  If it helps, I'm running Ubuntu 9.04.