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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0answers
8 views

Another form of the Sandwich theorem (for derivatives in dimension $1$)

Here is the theorem : "Let $I\subseteq \mathbb{R}$ an interval which contains $a\in \mathbb{R}$. Let $M$ and $m$ two functions defined on $I$, differentiable at $a$ and $f$ a function defined on $I$ ...
1
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0answers
28 views

base change of exterior powers

Let $n\geq 0$ be an integer, $R\to R'$ a ring homomorphism, and $M$ an $R$-module. Then the following holds: $$\bigl(\bigwedge^n_R M\bigr)\otimes_r R' \cong \bigwedge_{R'}^n\, (M\otimes_r R').$$ I ...
0
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1answer
24 views

Proof of the Curtis-Hedlund Theorem: Why is there a function $\mu\colon A^S\to A$ such that $\tau(x)(1_G)=\mu(x_{|S})$ for all $x\in A^G$?

Here is the Curtis-Hedlund Theorem and its proof [the sets $V(\cdot,\cdot)$ used in this proof are explained below.]: My problem is I am not sure that I have understand that correctly. So I ...
2
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1answer
56 views

Question on the Squeeze theorem

In this theorem we consider the functions $f$, $g$ and $h$ which are defined on $\bar {\mathbb{R}}$ except possibly at $a \in \bar {\mathbb{R}}$ but could we have the limit in $a$ of these three ...
0
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1answer
25 views

Which of the following conditions must necessarily be true?

Suppose that $\{A, B\}$ is a set of mutually exhaustive conditions, and that $\{C, D\}$ is another set of mutually exhaustive conditions. If the following implications are true: $$A \Longrightarrow ...
3
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1answer
42 views

Motivation for constructing $F$ s.t. $\ker(\text{curl}) \subset \text{Im}(\text{grad})$, $\ker(\text{div}) \subset \text{Im}(\text{curl})$

In 'from calculus to cohomology', we consider the space $V$ of smooth functions $U \to R^3$, with $U \subset R^3$ star-shaped (i.e. convex), and for cohomology reasons (showing $H^1(U)=H^2(U)=0$) we ...
4
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2answers
42 views

Upperbound this difference between two log expressions

I have the difference between the following log expressions and I am trying to bound the difference, $$F= \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) - \log \left(1+ ...
2
votes
1answer
79 views

Inequality.such as Nesbitt

Let $a,b,c >0 $ , prove that: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} \leq \dfrac{a^2}{b^2+c^2}+\dfrac{b^2}{c^2+a^2}+\dfrac{c^2}{a^2+b^2}$$
2
votes
1answer
30 views

Shuffling cards and laying them out in order

The numbers from 1 to 50 are printed on cards. The cards are shuffled and then laid out face up in 5 rows of 10 cards each. The cards in each row are rearranged to make them increase from left ...
2
votes
6answers
73 views

Discrete Mathematics: $mn + 2m + 2n + 2 = n$ proof of uniqueness of $m$, $\forall n \in \mathbb{Z}$

Prove: There exists a unique integer $m$ such that for every integer $n$: $$mn + 2m + 2n + 2 = n$$ However I am not sure if my proof is correct. How do I prove uniqueness of $m$? I prove it by ...
2
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3answers
52 views

Tips on constructing a proof by induction.

So right now I'm working on a discrete mathematics course and I've been having a bit of trouble figuring out how to prove certain equations using mathematical induction. I have very little trouble ...
1
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0answers
38 views

proof that the expression is Real for any $z$

Please help me with this problem, I'm clueless here. $\ \ \ \ (\bar{z}+1-2i)^{1985} + (\bar{z}+1+2i)^{1985}$ $\ \ \ \ $proof that the expression is Real for any $z$
6
votes
2answers
36 views

Question about loss of generality in proofs

My concern is with choosing specific conditions within a proof to arrive at a general result. As an example, I'll use the proof that $\mathbb{Q}$ is dense in $\mathbb{R}$. The proof I know goes as ...
1
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1answer
31 views

Stuck on basics: How to prove that {subst($\alpha$,s)} is well defined?

So I feel like this is a really basic point that I'm missing and I can't really manage to prove that: So I have a substitution function $s: Var \rightarrow WFF$ and a subst function: $WFF \times ...
0
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2answers
57 views

Induction proof for natural numbers

I am unable to proceed with the below claim. $$2^{m} \times 2^{n} = 2^{m+n}$$ Could anyone let me know how to prove the above claim using induction proof? I was able to derive proof for odd natural ...
4
votes
1answer
64 views

A combinatorial proof of Wilson's Theorem

I am looking for a combinatorial proof of Wilson's Theorem. Something along the lines of this kind of proof. $\textbf{Combinatorial proof of Fermat's Little Theorem}$ First consider a $p$ -tuple and ...
5
votes
4answers
235 views

For every positive integer $n, n^2 + 4n + 3$ is not a prime

Prove: For every positive integer $n, n^2 + 4n + 3$ is not a prime. I tried to disprove the statement, which I could not using several number examples with constructive proof. However I am not sure ...
1
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5answers
76 views

If $a,b\in\mathbb R$ with $a<b$, then there is some rational $r$ with $a<r<b$. [duplicate]

How do you prove this question? I was thinking proving contrapositive. But I was stuck..Thanks guys.
1
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2answers
71 views

Clarification regarding Drinker's paradox [duplicate]

This is the informal proof of Drinker's paradox The proof begins by recognising it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one ...
2
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3answers
115 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
1
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2answers
39 views

methods of proof, discrete mathematics

"Disprove: For all integers $r, m,$ and $n$, if $r$ divides $mn$ then either $r$ divides $m$ or $r$ divides $n$." I am not sure if I am on the right track To disprove I try the negation of a ...
3
votes
1answer
47 views

Prove that a square of a positive integer cannot end with $4$ same digits different from $0$

Prove that a square of a positive integer cannot end with $4$ same digits different from $0$. I already proved that square of positive integer cannot end with none of digits $1,2,3,5,6,7,8,9$ using ...
3
votes
3answers
59 views

Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$

The problem does not say it but I think solutions should be from $\mathbb{R}$. I tried to express the left sum as a sum of squares but that does not work out. Any suggestions?
1
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3answers
59 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
0
votes
1answer
18 views

Conditions so that Lebesgue Covering Dimension and “Usual” Dimension are Equal

The definition of covering dimension is as follows: The ply of a cover is the smallest number $n$ (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement ...
1
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0answers
34 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
0
votes
1answer
41 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
1
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1answer
25 views

Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
2
votes
3answers
68 views

Injective Ring Homomorphism

I seem to be having the wrong impression of what $p$ stands for; is $p(x)=x(x+1)(x+2)$ or is it something else? Clarification would be appreciated so that I can complete the lemma below. Consider ...
3
votes
1answer
50 views

Proving $P$ by proving $\neg Q$ and knowing $P\lor Q$

This may sound silly. I used to remember studying this in physics class and I thought of asking it in physics.stackexchange and then later I decided to ask it here itself. Let's say, under some ...
0
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0answers
39 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
1
vote
2answers
65 views

How to find the limit of the sequence given by $X_{n+1} = \frac12 X_n + \frac1{X_n}$

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1{X_n}$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still ...
0
votes
1answer
39 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
6
votes
1answer
71 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
1
vote
3answers
31 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
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votes
4answers
51 views

Prove that lim of x/(x+1) = 1 as x approaches infinity

I want to prove that $$\lim\limits_{x\to \infty} \frac{x}{x+1}=1$$ I know that I need to show that: $$\left|\frac{1}{x+1}\right| \lt \epsilon$$ But I'm not sure how to manipulate it. Any help or hint ...
1
vote
0answers
37 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
8
votes
3answers
188 views
+50

Examples of useful, insightful, and interesting hand-waving

It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers ...
6
votes
2answers
111 views

How to explain to a layman why Fermat's Last Theorem involves non-trivial math?

Fermat's Last Theorem states, given$$x^n + y^n = z^n$$ no three integers $x,y,z$ will satisfy the equation given integer value of $n$ greater than two. On the surface this seems like something that ...
4
votes
6answers
412 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
2
votes
5answers
124 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
0
votes
1answer
19 views

Proof if $n_k < n_{k+1}$ for all $k \in \mathbb{N}$, then $n_k \geq k$ for all $k \in \mathbb{N}$.

So if we proceed by induction on $k$, the base case $k = 1$ works since $n_1 \geq 1$ is true because $1$ is the smallest integer in $\mathbb{N}$. For the induction hypothesis, we have that $n_k \geq ...
0
votes
1answer
15 views

Deduce an inequality by using Bernoulli's Inequality

Deduce $c^n\geq c \forall n\mathbb\in{N},c>1$ What I have tried is $$\text{Let }x=c-1$$ Then I substitute it into the Bernoulli's inequality, that is $$c^n\geq1+n(c-1)\geq 1+nc-n\geq nc+1$$ How ...
0
votes
3answers
17 views

Proving a surjective function by given property

Suppose $f:E\rightarrow F$ and for any $A\subset F,A=f(f^{-1}(A))$. Show that f is surjective. What i have tried is $$\text{Let }y\in A $$ $$\{y\}\subset A$$ $$\{y\}= f(f^{-1}{\{y\})}$$ And i stuck ...
-1
votes
4answers
49 views

Counting candies in boxes

There are $5$ boxes containing $80$ candies. After taking $\frac{1}{5}$ of the candies in the first box and putting them in the seconf one, we take $\frac{1}{5}$ of the candies in the second box and ...
0
votes
0answers
52 views

Don't understand proof that if $\{x_n\}$ is Cauchy and if some $x_{n_k} \rightarrow x$, then $x_n \rightarrow x$

So by definition of Cauchy, for all $\epsilon > 0$ and $i, j \in \mathbb{N}$, there exists an $M$ such that for all $i, j \geq M$, then $|x_i - x_j| < \epsilon'/2$ if we let $\epsilon = ...
0
votes
1answer
41 views

Don't understand proof that convergence implies Cauchy

So we are given that $x_n \rightarrow x$, so we can let $\epsilon = \epsilon'/2$ and there definitely exists an $N$ such that for all $n \geq N$, $|x_n - x| < \epsilon'/2$. Also by the triangle ...
2
votes
2answers
47 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
0
votes
1answer
42 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
1
vote
2answers
21 views

If $\sup A = 5$ and $B = \left\{ 3a \mid a \in A \right\}$ then $\sup B = 15$

Prove that if $A \subset \mathbb{R}$, $\sup A = 5$, and $B = \left\{ 3a \mid a \in A \right\}$, then $\sup = 15$. I tried to do contradiction by assuming the hypothesis and that there is a number ...