For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0
votes
2answers
29 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

I am hoping a few of you mathematicians more experienced with writing proofs might give me some guidance here and possibly give me some ideas about how to restructure the following into a more ...
0
votes
4answers
53 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
-2
votes
2answers
32 views

Proof Techniques ( Soft Question )

I've been googling around for books of methods of mathematical proofing, and I haven't had much luck finding anything reputable in book form. I do recall running by a few in a university library ( I ...
2
votes
2answers
34 views

Check correct delta in eps-delta proof

I been stuck now with this seemingly simple exercise for some time. I need to show that: $|x^2-4| < \epsilon$ when $0 < |x-2| < \epsilon(5+\epsilon)^{-1}$ But I'm at a loss. I know that I ...
0
votes
2answers
46 views

Complex Analysis Problem and Advice

Let $f$ be an odd function that is holomorphic in $\mathbb{C}- \{0\}$ such that $|f(z)| \leq \dfrac{1}{|z|}+ |z|^2, $ where $z \neq 0.$ Could someone advise on how to show $f(z) = \dfrac{a_{-1}}{z} + ...
-2
votes
1answer
55 views

Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$

I am unsatisfied with the answers here. (Half of which used algebraic methods despite being advised not to!) Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ ...
2
votes
1answer
219 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
13
votes
16answers
1k views

Beautiful, simple proofs worthy of writing on this beautiful glass door [on hold]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
2
votes
1answer
91 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
0
votes
1answer
49 views

Show that $f^{[n]}(0)=0$ for all $n=0,1,2…$

Let $$f(x)=\left\{ {\matrix{ {{e^{ - {1 \over {{x^2}}}}},x \ne 0} \cr {0,x = 0} \cr } } \right.$$ Show that $f^{[n]}(0)=0$ for all $n=0,1,2,\cdots$ The proof: First note that for ...
2
votes
5answers
397 views

Help with a proof: Given $a\ne 0$ and $b\ne 0$ and $a \lt \frac{1}{a} \lt b \lt \frac{1}{b}$. Prove $a\lt -1$

This question is from the book How to Prove It and I'm having trouble getting started with it. The book provides the hint "first prove $a \lt 0$". However, I can't figure out how to get that far with ...
0
votes
0answers
15 views

Ingredients for Proving that a set is bounded below

I am having a hard time to really understand the definition of bounded below. The definition states : Let S be a nonempty subset of real numbers, we say S is bounded below if there is a c ∈ ℝ ...
1
vote
2answers
43 views

What are the entire functions $f$ such that $|f'(z)| \leq |f(z)| \ ? $

Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$ Hints will suffice, thank you.
0
votes
1answer
52 views

Evaluation of $\begin{align} \int^{\infty}_{0}\end{align} \dfrac{1}{1+x^n}dx$ with the use of Residue theorem [duplicate]

Could anyone advise me on how to show$\begin{align} \int^{\infty}_{0}\end{align} \dfrac{1}{1+x^n}dx=\dfrac{\pi}{n\text{sin}\dfrac{\pi}{2}} ,\ $ for all integers $n \geq 2 \ ?$ Thank you. Here is my ...
-4
votes
0answers
17 views

Let (A,≼A) and (B,≼B) be partially ordered sets. [duplicate]

Let (A,≼A) and (B,≼B) be partially ordered sets. Define C = A×B and define the relation ≼' on C by (a,b)≼'(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′). (a) Prove that ≼' is a partial order on C. (b) Prove that if a ...
0
votes
1answer
40 views

Let m ∈ N. Define the relation ≡^ on Z by a ≡^ b for a, b ∈ Z if and only if a ≡ ±b (mod m).

(In other words, the relation ≡^ holds if either a ≡ b (mod m) or a ≡ −b (mod m).) Prove that the relation ≡^ on Z is transitive. ======= I believe there are 3 properties that it must meet ...
2
votes
2answers
229 views

Prove that f(x)=1/(1+x) is not uniformly continuous

How can I prove that $f(x) = \dfrac{1}{1+x}$ is not uniformly continuous on $(−1,\infty)$. Thank you.
1
vote
2answers
56 views

Trigonometry and triangle proof

Question: Prove that in an acute angle triangle ABC: $$\tan A\tan B +\tan A \tan C + \tan B \tan C \geq 9$$ I have no idea where to even begin this question. Please help me!
1
vote
1answer
20 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
0
votes
1answer
16 views

Supremum of a subset is less or equal than infimum of another subset

Let X,Y be two bounded subsets of R satisfying the following proposition 1 : $\forall x \in X, \forall y \in Y ( x \leq y ) $ I wanted to know if there's a direct proof of sup X $\leq$ inf Y. I ...
0
votes
2answers
115 views

Any proof that verify why the limit of the difference is the difference of the limits?

I did a research on internet and books about why the difference of the limits is the difference of the limits, but i didn't get any result of this proof. I would appreciate if somebody can help me. ...
1
vote
1answer
34 views

Complex Analysis -Proving convergence

Suppose that $$z_n,z\in G:=\mathbb{C}-\{z\,:\,z\leq 0\}$$ and $$z_n=a_n e^{i\theta_n},z=ae^{i\theta}$$ where $-\pi<\theta,\theta_n<\pi$. Prove that if $z_n\to z$ then $\theta_n\to\theta$ and ...
2
votes
1answer
38 views

An inequality for a quotient of polynomials

I am trying to prove the following to be true for $n > 1$: $$\frac{n^4}{n^3 + 1} \le Cn$$ It seems like there is some basic rule where you multiply the 1 in the denominator by a value which makes ...
2
votes
1answer
216 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
0
votes
3answers
237 views

How do I solve delta epsilon proofs for quadratic equations?

For the $\lim\limits_{x\rightarrow 2} (x^2 + 5x - 2) = 12$ I need to show how to find a $\delta$ such that $|f(x) - L| < \varepsilon$ for all $x$ satisfying $0 < |x - a| < \delta$. Help is ...
2
votes
2answers
33 views

How to find all values of $z$ at which $\begin{align} \sum^{\infty}_{n=1} \dfrac{1}{n^2} \end{align}\text{exp}\left(\dfrac{nz}{z-2}\right)$ converges

Could anyone advise me on how to find all $z$ such that $\begin{align} \sum^{\infty}_{n=1} \dfrac{1}{n^2} \end{align}\text{exp}\left(\dfrac{nz}{z-2}\right)$ converges ? Does it suffice to find all $z$ ...
0
votes
2answers
131 views

Why the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 is 1? [duplicate]

I need a rigorous proof that verify why the limit of $\dfrac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$. I tried before but i do not know how start this proof. I would appreciate if somebody help me. ...
4
votes
4answers
88 views

If $f,g$ are entire functions such that $f(g(z))=0, \forall z, $ then $g$ is constant or $f(z) =0, \forall z \ ?$

Let $f,g$ be entire functions such that $f(g(z))=0, \forall z.$ Could anyone advise me on how to prove/disprove: either $g(z)$ is constant or $f(z) =0, \forall z \ ?$ Hints will suffice, thank you.
0
votes
2answers
87 views

How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$

I am taking a real analysis course. I have the following statement: Prove that if $a$ belongs to $\mathbb R$, such that $0\leq a < \epsilon$, for all $\epsilon > 0$, then $a = 0$ I ...
1
vote
1answer
115 views

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
0
votes
2answers
41 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
6
votes
1answer
867 views

Proof Involving a Problem from “Good Will Hunting”

I don't know if any of you have seen the movie "Good Will Hunting" but there is a particular mathematics problem in the movie that is of interest to be. One of the problems used in the movie is "Draw ...
2
votes
1answer
47 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
0
votes
2answers
34 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
1
vote
1answer
46 views

'Identity theorem' for Meromorphic functions

If $f_1,f_2$ are meromorphic functions in $D$ and there exists a sequence of pairwise distinct points $z_n \in D$ such that $z_n \to z_o \in D$ and $f_1(z_n)=f_2(z_n),$ then $f_{1} \equiv f_2$ on $D.$ ...
0
votes
2answers
30 views

Convergence of integral, that is absolutely convergent, proof

Can you think of any proof on convergence of improper integral, that is absolutely convergent? It is so obvious, that I really don't know where to start. Triangle inequality gives us ...
2
votes
1answer
29 views

Radius of convergence of sum of complex power series

Could anyone advise me on how to find radius of convergence of $\sum^{\infty}_{n=1} [\frac{1}{n^2}+(-2)^n]z^n \ ?$ Thank you. My attempt: radius of convergence of $\sum^{\infty}_{n=1} ...
0
votes
1answer
27 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
4
votes
0answers
36 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
52
votes
8answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, ...
1
vote
2answers
50 views

Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
1
vote
0answers
13 views

Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
0
votes
0answers
18 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
2
votes
2answers
69 views

Exercise about truth functions in J.R.Shoenfield's “mathematical logic”

The first exercise in Joseph R. Shoenfield's "mathematical logic" is: An n-ary truth function $H$ is definable in terms of the truth functions $H_1,\dots,H_k$ if $H$ has a definition ...
0
votes
0answers
8 views

Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
4
votes
1answer
1k views

Integrating $\int \sin^n{x} \ dx$

I am working on trying to solve this problem: Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$ Here are the steps that I ...
2
votes
4answers
73 views

Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$? [duplicate]

I have tried this question so hard but still stuck here. It seems like easily provable if all $n$ are all positive numbers but in this question, the $n$ is bigger than $1$. original question : prove ...
2
votes
2answers
369 views

Dijkstra's Shortest-Path Algorithm

I'm presented with the following algorithm: Dijkstra's Shortest-Path Algorithm This algorithm finds the length of a shortest path from veftex $a$ to vertex $z$ in a connected, weighted ...
0
votes
2answers
95 views

Does there exists an entire function with the following property: $f\left(\frac{1}{n}\right)= \frac{n^4}{1+n^4}, n =1,2,…$

Could anyone advise me on how to use the Identity theorem to determine whether there exists an entire function with the following property: $f\left(\dfrac{1}{n}\right)= \dfrac{n^4}{1+n^4}, n =1,2,...$ ...
-3
votes
1answer
65 views

why is (a proof of 1+1=2) necessary? [closed]

http://mathforum.org/library/drmath/view/51551.html Now, empirically speaking, I have this one on the left, and that one on the right, and together they make these two right there. why is it ...