# Tagged Questions

If you already have a proof for some result, but want to ask for a different proof (using different methods).

42 views

### Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the parabola ...
36 views

36 views

### In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
60 views

144 views

### Proof of a statement about eigenvalues and eigenvectors.

How can i proof the following: Let $\mathbb L: V\rightarrow V$ be a linear mapping. Let $v_1,v_2,..,v_n$ non-zero eigenvectors with eigenvalues $c_1,c_2,..,c_n$ respectively, also let the ...
77 views

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, $$f(A_1)+f(A_2)+f(A_3)... 1answer 17 views ### How to proof that the set of all X such that X.A{\ge} c to some real number c is convex? How can i proof the following statement: " Let \mathrm A\in \mathbb R^{n} and \mathrm c\in \mathbb R, the set \mathbb S of all elements belonging to \mathbb R^{n} and satisfying the ... 1answer 60 views ### Show that \int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)} for \mathbb{N}\ni n\geq 2 Show that$$\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$$for \mathbb{N}\ni n\geq 2. Let S=\{r\mathrm e^{\mathrm i\varphi}\in\mathbb{C} \mid 0\leq r\leq R,0\leq \varphi\... 1answer 83 views ### Obscure proof that + and \times are continuous? I am looking for proof of + and \times are continuous operations without using the standard definition of continuity (1. \epsilon-\delta, or 2. preimage of open sets or 3. sequential ... 1answer 32 views ### Slicker way to prove \rho is a metric Let d be a metric on X, and define \rho: X^{2} \to \mathbb{R} as$$\rho(x,y)=\frac{d(x,y)}{1+d(x,y)}$$The difficulty is in checking the triangle inequality. So, I can prove this by writing f(t)... 1answer 22 views ### Sequence of continuous function converge uniformly to continuous function Theorem: Let (f_n) be a sequence of continuous functions. If (f_n) converge uniformly to f, then f is continuous. I don't like the Rudin's approach, then I try to do my own way, but it seems ... 0answers 27 views ### Shortest proof for: if f is continuous in an interval A then \{x,y\in A:f(x)=f(y)\land x\neq y\} is empty or uncountable [duplicate] Prove that a continuous function on an interval A have zero or uncountable points such that f(x)=f(y) for x\neq y\in A. I did a proof that I will trace briefly: When \{x,y\in A:f(x)=f(y)\land ... 1answer 105 views ### How to integrate \int{1\over \sqrt{x^2-1}}\mathrm d x another technique without use trigonometry How can I integrate \int{1\over \sqrt{x^2-1}}\mathrm d x without using trygnometry? I mean using other methods like substituition, integration by parts (I tried these two but I think I am not seeing ... 4answers 111 views ### Prove \ \sin(x) < x \ \ \ \forall x \in(0, 2\pi) Problem : Prove \sin(x) < x \ \ \ \forall x \in(0, 2\pi) Now I have a possible solution for this, using limits and the first derivatives of \sin(x) and x, but I don't feel it's a very ... 5answers 688 views ### Proving \sum_{k=1}^n k\cdot k! = (n+1)!-1 without using mathematical Induction. [duplicate] Possible Duplicate: Summation of a factorial This equation is given:$$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$I've solved it using mathematical induction but I'... 1answer 58 views ### How to integrate \frac{dx}{(x^2+k^2)^m}, with m positive integer. How can I integrate:$$\int \frac{1}{(t^2+k^2)^m}\, dt$$without trygonometric substituition? where t= (x+(p/2)) and k= (1-(p^2/4)) coming from an equation with complex roots: x^2 + px + q. ... 1answer 88 views ### Alternate proof of the integral: \int_0^1 x^x(1-x)^{2x}\,dx\neq3/8 I am looking into the integral:$$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$How might you prove this to be true? What's tough is that the integral$$3/8\lt I<0.37503$$numerically. I managed ... 1answer 59 views ### Proof of Tychonoff's Theorem for an undergrad In the midst of learning about compactness I come across Tychonoff's Theorem: Let \{X_i : i \in \mathcal{A}\} be any collection of compact spaces. Then \displaystyle\prod_{i \in \mathcal{A}}X_i... 1answer 68 views ### Elegant proof of an elementary result in Linear Algebra I've been reading Hoffman Kunze, and I came across this theorem (theorem 9) which has a long and tedious proof. I've been wondering wether there could be a more elegant proof. Theorem 9. Let e ... 2answers 894 views ### A simply-connected closed surface is a sphere From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that S^2 is the only closed surface with trivial \pi _1. That's easy because the fundamental group ... 0answers 56 views ### Shortest proof for showing \mathbb{Z}[\frac{1+\sqrt{-19}}{2}] is a PID. I'm looking for an easy proof for that \mathbb{Z}[\frac{1+\sqrt{-19}}{2}] is a PID. One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it ... 2answers 25 views ### Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces. I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ... 0answers 35 views ### Proof by induction of Gronwall's inequality I've an exercise which is the following: Gronwall’s Inequality Let A > 0, B \geq 0. Let (\epsilon_j)_{j \in \mathbb{N}} be a sequence of real numbers with$$|\epsilon_{j+1}| \...
I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
(Context: I'm self-studying Murray (1984). I learned (and have forgotten quite a lot of) real and complex analysis. I'm willing to relearn and to look up references.) Problem: if $f=O(g)$, show that ...