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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1
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0answers
66 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
0
votes
1answer
112 views

Show that an entire function that is real only on the real axis has at most one zero, without the argument principle

Could someone advise me on how to approach this problem: Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. without the use of argument principle ...
1
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2answers
98 views

The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.

I'm trying to prove the following claim: Let $F\colon M \to N$ be a differentiable application beetween $C^\infty$ manifolds. Then the differential $\text dF_p\colon T_p M \to T_{F(p)}N$ is ...
1
vote
2answers
77 views

If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $

Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
1
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2answers
341 views

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

Thanks for the help! I've written a more detailed proof. The hints were great.
-1
votes
2answers
67 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 ...
0
votes
5answers
85 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
1answer
47 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 ...
-1
votes
1answer
100 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 $k {n \choose k } = n {n - 1 \choose k -1}$ ...
0
votes
2answers
64 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} + ...
0
votes
1answer
82 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 ...
15
votes
16answers
2k views

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

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 ...
1
vote
2answers
63 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
136 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 ...
0
votes
1answer
49 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 ...
1
vote
2answers
65 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
374 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
269 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 ...
2
votes
1answer
45 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
47 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 ...
3
votes
2answers
50 views

How to find all values of $z$ at which $\sum_{n=1}^{\infty} \frac{1}{n^2} exp(\frac{nz}{z-2})$ 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
146 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. ...
0
votes
2answers
185 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. ...
0
votes
2answers
123 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 ...
4
votes
4answers
164 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.
2
votes
1answer
118 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
2answers
127 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 ...
0
votes
2answers
74 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
105 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 $D.$ ...
1
vote
2answers
43 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 ...
3
votes
1answer
93 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
29 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. ...
1
vote
0answers
28 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 ...
4
votes
1answer
96 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 ...
0
votes
1answer
38 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 ...
0
votes
0answers
50 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 $w_i \neq z_j$ ...
1
vote
2answers
59 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 ...
2
votes
2answers
131 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 ...
2
votes
4answers
135 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 ...
1
vote
0answers
34 views

Proof Verification: For $f: A \to B$ and $T \subset B$ show that $f^{-1}(T') = (f^{-1}(T))'$.

I want to know if my proof is correct and if the strategy I use should be used for all questions of this form. Compliments are taken with respect to the set $B$. My method of proof would be to first ...
0
votes
2answers
128 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: $f\left(\dfrac{1}{n}\right)= \dfrac{n^4}{1+n^4}, n =1,2,...$ ...
2
votes
1answer
155 views

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
1
vote
1answer
30 views

If $(y_n(x))_{n \in \mathbb{N}}$ is uniformly convergent, so is $(f(x,y_n(x)))_{n \in \mathbb{N}} \ ?$

Let $f$ be a continuous function defined on $[a,b] \times [c,d]. $ Consider $(y_n(x))_{n \in \mathbb{N}}$ such that it is uniformly convergent on $[e,f] \subseteq [a,b].$ Could anyone advise me on ...
1
vote
2answers
70 views

Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$. The proof is easy enough ...
4
votes
1answer
66 views

To show that a concretely defined group is isomorphic to an explicitly presented group, what strategies are available?

I have a homework problem of the following form. We're given presentation of a group $\langle x,y \mid R\rangle$ explicitly, and two matrices $X,Y \in \mathrm{GL}(\mathbb{C},2).$ We know $X$ and $Y$ ...
2
votes
1answer
92 views

Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+…+x+1)$

So what I am trying to prove is for any real number x and natural number n, prove $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$ I think that to prove this I should use induction, however I am a bit stuck ...
1
vote
1answer
62 views

Proof using vectors - trigonometric formulas

Question: If two vectors a and b make angle $\alpha$ and $\beta$ with the x-axis, prove, using vectors, that: $$\cos(\beta - \alpha) = (\cos \alpha) (\cos\beta) + (\sin\alpha) (\sin\beta)$$ I tried ...
4
votes
4answers
80 views

Proof by contradiction that $(n+1)^3 \not= n^3 +(n-1)^3$ for $3$ consecutive positive integers

Prove by contradiction that if n-1, n, n+1 are consecutive positive integers, then the cube of the largest cannot be equal to the sum of the cubes of the other two. Assume that: $$ (n+1)^3 = ...
2
votes
1answer
68 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
75
votes
11answers
3k views

Problems that become easier in a more general form

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...