For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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0
votes
2answers
62 views

How would this problem need the Mean Value Theorem?

I'm asked to square the inequality and use the Mean Value Theorem to prove that $$\sqrt{1+x} < 1 + \frac{x}{2}$$ for $x>0$. Unfortunately, I don't really understand why I would need such a ...
2
votes
2answers
44 views

Free and bound variables in “if” statements

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case ...
0
votes
0answers
26 views

U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
2
votes
1answer
44 views

Proving a theorem on limits

I need to prove that: If $$\lim_{(x,y) \to (a,b)}f(x,y)= 0 \text{ and } g(x,y)\leq k,$$ then: $$\lim_{(x,y) \to (a,b)}f(x,y) g(x,y) =0.$$ My approach is like follow: ...
3
votes
2answers
80 views

Idea of a proof by contradicton

Is the idea of a contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the conclusion)? Or ...
0
votes
0answers
19 views

complete logic for proving inequalities

Last semester I took a course on algorithm analysis a big part of which was proving that the running time function of a program was in the set $O(f(x))$ for some $f$. To prove $f\in O(g(x))$ one ...
1
vote
2answers
41 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
74 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
4
votes
3answers
98 views

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
1
vote
2answers
135 views

Is there a specific name for a corollary of a conjecture?

How do you call a corollary of a conjecture? Is there a specific name for it? Can it be called simply 'corollary'? Can't it be called 'corollary'? I mean, does the label 'corollary' imply that the ...
1
vote
0answers
33 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
2
votes
0answers
39 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
0
votes
2answers
50 views

Suppose $F$ and $G$ are families of sets.

Suppose $F$ and $G$ are families of sets. Prove that $\bigcup F$ and $\bigcup G$ and are disjoint iff for all $A∈F$ and $B∈G$ , $A$ and $B$ are disjoint. It has been suggested to use contrapositive ...
-1
votes
3answers
112 views

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
2
votes
2answers
32 views

Proofs involving Well-Defined and One-to-One

Chartrand, 3rd Ed, P224-225: Define a relation $R$ as a relation from A to B. $R$ is well-defined means: $(a,b), (a,c) \in R \implies b = c$. P220: A function $f: A \to B$ is one-to-one ...
3
votes
1answer
50 views

$f'$ strictly increases and $f'(c)=0$. There exist $x_1 < c < x_2$ such that $f'(c)=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$

Question: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Assume that $f'$ is strictly increasing. Show that for any $c\in(a,b)$ such that $f'(c)=0$, there exist $x_1, x_2 \in [a,b], ...
2
votes
0answers
62 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
2
votes
2answers
36 views

Prove by induction that $\sum_{i=1}^n i!\times i=(n+1)!-1$ for all $n\in \mathbb{N}$

So far I have, If $P(n):\sum_{i=1}^n i!\times i=(n+1)!-1$, then $P(1):\sum_{i=1}^1 i!\times i=1$ and $(1+1)!-1=1$ , so P(1) is true. I know I now have to assume P(K) is true, such that ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
2
votes
3answers
44 views

Strong Induction, assuming k<n where k and n are not numbers

In strong Induction for the induction hypothesis you assume for all K, p(k) for k If for example I am working with trees and not natural numbers can I still use this style of proof? For example if I ...
2
votes
1answer
47 views

Proof involving maximum weight of edge in minimum spanning tree

Let $G$ be a minimum spanning tree of a complete graph. Let $e$ be the maximum weight edge in $G$. I'd like to proof that given any other spanning tree $G'$ of this graph, being $j$ the maximum weight ...
1
vote
1answer
25 views

How to prove a bijection?

I know what a bijection is and how to prove it when given a function, but how to do it when you are only given sets.
3
votes
1answer
63 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...
0
votes
1answer
29 views

Converse of DIC

I am looking to prove the converse of the Divisibility of Integer Combination. I know how to prove the contrapositive of this statement but not the converse ... any help? Below is my attempt at ...
0
votes
2answers
51 views

Equivalence relations and equivalence classes

I dont know how to start this proof? Also, our professor did not explain equivalence classes fully so I am not understanding them very well.
1
vote
0answers
21 views

Is cocycle condition necesarry for coassociativity of coproduct and why is it called “cocycle condition”?

Let $\Delta_0$ and $\Delta$ be coproducts related by \begin{equation} \Delta h = \mathcal F \Delta_0 h \mathcal F^{-1} \end{equation} where $\mathcal F \in H\otimes H$ is Drinfeld twist and $h \in H$ ...
0
votes
0answers
32 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
2
votes
1answer
48 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
3
votes
1answer
61 views

Prove $\neg\exists x \neg P(x)$ is a logical consequence of $\forall x P(x)$

How would you prove that 2. is a logical consequence of 1. using a Fitch style proof? $\forall x P(x)$ $\neg\exists x \neg P(x)$
0
votes
1answer
47 views

Proof for binary tree is a planar graph

Suppose G is a binary tree. Is G necessarily planar? Give a proof, or a counterexample. My guess is that it is indeed planar but I am struggling to find a formal proof for this. EDIT: Is there a ...
0
votes
2answers
27 views

How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
2
votes
2answers
49 views

Let f be a continuous real valued function on R, and prove that $f(x)$ is constant using the fourier series.

I missed my class where we went over the fourier series and am having extreme issues with this homework question. $f(x) = f(x + 1) = f(x + \sqrt{2})$ Is there anyone who could be kind enough as to ...
3
votes
2answers
44 views

equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H $. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
1
vote
1answer
126 views

Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.

I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers. A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers ...
3
votes
3answers
39 views

Proof that for every non empty $A \subset \mathbb{N}$ there's a $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f \circ f =f$ and $f(N)=A$

I need to prove that, for every non empty $A \subset \mathbb{N}$, there's a $f: \mathbb{N} \rightarrow \mathbb{N}$ with $f \circ f =f$ and $f(\mathbb{N})=A$ Unfortunately I have no idea where to ...
0
votes
2answers
136 views

Is this a mathematical statement?

Is this a mathematical statement: Suppose this statement is false. I know what a mathematical statement is: it's either true or false. But the suppose is what's confusing me.
0
votes
1answer
45 views

Help with proof for solving an ODE using contraction mapping theorem

I'm trying to follow a proof for solving the ODE $$\frac{df}{dx} = (f(x)+x)x$$ For $0 \leq x \leq 1$ with the initial condition $ f(0)=0$. The proof I am following goes like this Define ...
38
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
6
votes
1answer
105 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
0
votes
1answer
32 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
3
votes
5answers
92 views

Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
3
votes
2answers
87 views

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
3
votes
2answers
60 views

Is this a proof by contradiction?

Below is a proof that any group of order $p^2$ is abelian $(p$ prime of course). Let $Z \left({G}\right)$ be the center of $G$. We know $|Z(G)|>1$. $\color{blue}{\text{Suppose}} \left\vert{Z ...
3
votes
1answer
123 views

Why is this more-detailed proof more acceptable than its trivial counterpart?

Say that we're asked to give a proof of 'proof by induction'. i.e. for some property $P$, proving that $$\forall n,P(1) \wedge [P(k) \implies P(k+1)] \implies \forall n, P(n)$$. Now, I understand ...
2
votes
2answers
31 views

Proving a theorem about the limit of a function

The theorem is as follows: $$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$ ...
2
votes
2answers
37 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
0
votes
1answer
26 views

Proof homomorphism between graphs

Given two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, an homomorphism of $G_1$ to $G_2$ is a function $f:V_1 \rightarrow V_2$ such $(v,w) \in E_1 \rightarrow (f(v),f(w)) \in E_2$. We establish that ...
0
votes
1answer
26 views

Proof that a local minimum in a spanning tree is also a minimum spanning tree.

Be $G$ a connected graph with weights associated to its edges. Be $T(G)$ the graph that has the spanning trees of $G$ as vertex, and two spanning trees are adjacent to each other if and only if each ...
2
votes
2answers
21 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...