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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4
votes
1answer
30 views

Going from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨~q)

I am confused on how to go from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨ ~q). I know they are equal because I plugged them into a truth table and all of the rows have the same values. What would be some ...
2
votes
1answer
253 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
1
vote
1answer
43 views

Lattice theory question

I am having trouble with the following question Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$ My attempt: Let the ...
0
votes
1answer
20 views

What assumption is needed to add together equations in a vector space?

Suppose in a vector space $V$, we have $x+z=y+z$. We want to show that $x=y$. Note, by one of the vector space axioms, for all $z \in V$ there exists an additive inverse $v \in V$ such that for all ...
0
votes
3answers
86 views

Meaning of specific part of answer in delta-epsilon limit proof

In a math question I had such as the limit approaches $3$ in $x^4-x^2+1$ (help with formatting please I'm new), the answer was that $\delta$ equals $\left(1, \frac{\epsilon}{168}\right)$. In another ...
0
votes
1answer
85 views

Pigeonhole proof of the existence of two numbers with given sum [duplicate]

Let $|W|=m+1$ and $W$ be a subset of $X=\{1,2,3,\dots ,2m\}$ ($m$ is any natural number). Prove there exists two numbers in $W$ whose sum is $2m+1$. Can anyone give me a hint to prove this? I ...
2
votes
2answers
24 views

Basic Properties Explanation

In regards to divisibility I am having trouble wrapping my head around some of the concepts, more specifically some of the general properties of divisibility. for example, why is it possible for ...
1
vote
1answer
67 views

Application of Opening Mapping theorem

Let $f$ be a holomorphic function on open set $A$ such that $(Im(f(z))^3 + (Re(f(z))^4 =5.$ Could anyone advise me on how to use Open mapping theorem to prove $f$ is constant? Hints will suffice. ...
1
vote
1answer
76 views

Proving a graph is connected by minimum degree

Let number of vertices be >= 2. If the min degree of G >= n/2, then the graph is connected. I was trying to solve this using contradiction, but now I'm stuck. So I started out with "If the the min ...
0
votes
1answer
106 views

geometry circle proof

Use a common notion to prove the following result: If P and Q are any points on a circle with center O and radius OA then OP is congruent to OQ. Since O is the center and P & Q are any where on ...
0
votes
1answer
100 views

Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle ...
1
vote
1answer
239 views

Proof by cases. Formulate a conjecture. I don't get it. Question inside.

I don't understand this math question for my discrete math 2 class. FOrmulate a conjecture about the decimal digits that appear as the final decimal digit of the fourth power of an integer. Prove ...
4
votes
2answers
59 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
-1
votes
6answers
44 views

Proving $2^{2n}-1$ is divisible by $3$ for $n\ge 1$

So I decided to use induction. First, I started with my base case, $P(1) = 2^{2(1)}-1=3,$ so it's true. That means if $n = k$ is true, then $n = k+1$ is true also. So, $P(n+1)-P(n)$ would also be ...
15
votes
10answers
2k views

Having hard time understanding proofs by contradiction.

I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. $\textbf{Theorem:}$ If $P \rightarrow ...
6
votes
3answers
163 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
0
votes
0answers
122 views

Similar Triangles Proof - How to tackle proofs?

Well, I know it is repetitive.I have read the proof from different textbooks.But sometimes I feel doubtful about it all.Every time I try to prove it for myself, I fail at some points.I'm asking those ...
1
vote
3answers
36 views

A number to the group cardinality power

Well my question is how is possible this: Consider an element $g\in G$, where $G$ is a finite group, then you have: $g^{|G|}=e$ How can I prove it? Thank you.
3
votes
2answers
531 views

Proof of trigonometric identity using vector calculus

Question: Using vector calculus, show that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ I have no idea how to even attempt the question. A small hint to help me get started would be greatly ...
1
vote
2answers
630 views

An example of set with a countably infinite set of accumulation points

I have to give An example of set with a countably infinite set of accumulation points, and I say: We can consider the set or real numbers and we take an arbitrary real number $x$ then the interval ...
0
votes
0answers
54 views

LUB, GLB, maximum and minimum of a set .

I am not sure if my solution for the following problem is correct. Evaluate LUB, GLB, maximum and minimum (if they exist) of $\{-n: n ∈ \Bbb N\}$. My answers: LUB: $-1$ GLB: $-\infty$ max: $-1$ ...
2
votes
4answers
67 views

Exotic proofs of $\sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$

Let $p,n$ be positive integers. The following identity $\displaystyle \sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$ may be proved by induction or by successive uses of Pascal's rule (both ...
0
votes
3answers
57 views

Beginner Question about the definition of finite sets

Hi this question is in regard to a part of chapter 7 in Daniel Velleman's book "How to Prove it". I just began to learn about infinite sets and such and there is one part that confuses me. It starts ...
2
votes
1answer
58 views

Showing two functions are uniformly continuous

I have no idea how to prove this detail (uniformly continuous) about these functions because they're defined to $\infty$. I need the general mindset to prove it, or any ideas. Thanks in advance. $$ ...
0
votes
1answer
68 views

Proving that the exponential function is its own derivative, using the limit definition of $e$

I saw the proofs on the derivative of $\frac{d e^x}{dx}=e^x$ from here and the one that was intriguing was this : $$e^x:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...
1
vote
0answers
64 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
vote
2answers
88 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
74 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
vote
2answers
318 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
65 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
84 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
46 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
63 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
59 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
133 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
63 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
322 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
255 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
46 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
49 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
144 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
180 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
122 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
158 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.