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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1answer
48 views

Question about writing proofs for limit

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each $n \in \mathbb{N}$, let $a_n$, $b_n$ be real numbers. Also, let $a_{\infty}$, ...
0
votes
1answer
73 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
1
vote
1answer
35 views

Question about written proof for geometric summation

Suppose $\alpha$ $\ne$ $\beta$ $\in \{0, 2\}^\mathbb{N}$ Prove that $$\sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \ne \sum\limits_{k = 0}^\infty\frac{\beta(k)}{3^k}. $$ This is the written proof ...
0
votes
1answer
55 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
0
votes
1answer
44 views

How to prove that a given map is an injection?

Let $g:\mathbb{N_{m_1-1}}\rightarrow \mathbb{N}_{m_1}$, where: $$g(i) = \left\{ \begin{align} i & \text {, for } i<i_0 \\ i+1 & \text{, for } i \ge i_0 \end{align}\right.$$ and $i_0 ...
0
votes
5answers
53 views

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
1
vote
2answers
31 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
1
vote
4answers
46 views

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
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2answers
57 views
2
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1answer
50 views

Question about proof with geometric sums

I am confused on how to write proofs for geometric sums. I think that using the well ordering principle to find the least n $\in$ $\mathbb{N}$ with $\alpha(n)$ $\ne$ $\beta(n)$ would be a good ...
0
votes
1answer
48 views

Question about proofs with limits

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each n $\in$ $\mathbb{N}$, let $a_{n}$ be a real number. Also, let $a_{\infty}$ be a ...
0
votes
2answers
44 views

How do I reduce this induction problem at the k+1 step

Show that $2^n > n^2$ through induction and so far I got to the $k+1$ step, but I am stuck. I have $2^{k+1} = 2 +2^k$, but I don`t know how the book turned it into $k^2 +k^2$. The book then ...
0
votes
2answers
55 views

Proof by induction $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n [duplicate]

Prove by induction that $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n
0
votes
0answers
22 views

Orthonormality and fourier transform

If $g\in\mathcal{L}^2(\mathbb{R})$ then $\sum_{k\in\mathbb{Z}} |\hat{g}(\zeta+2k\pi)|^2=1$ for a.e $\zeta\in \mathbb{R} \Rightarrow \{g(.-k): k\in \mathbb{Z}\}$ is an orthonormal system. Please ...
1
vote
2answers
99 views

Prove: If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Can someone tell me if this proof is acceptable? Assume $A \not\subseteq C$. This means that there is an $x \in A$ such that $x \not\in C$. But since $\forall x \in A: x \in B$ and $\forall x \in B: ...
1
vote
2answers
74 views

Prove the reflexivity of $\subseteq$.

My professor gave me a list of exercises, I've been able to figure out what mechanism I should exploit to prove them, but I'd like to know if it's good. Until now we've been taught a little logic and ...
0
votes
2answers
34 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
0
votes
3answers
25 views

Can the proof for the following 4 cases be simplified to 2 cases?

Let $X$ and $Y$ be finite and disjoint sets. Suppose we are required to prove the following: $|X|\ge 0 \text{ and } |Y|\ge 0 \Rightarrow Q $ where $Q$ is some statement. Therefore, I know I need to ...
1
vote
1answer
63 views

How to prove that $f:\mathbb{N}\rightarrow X$ where $f$ maps to an element in a set, is a bijection?

Let $X$ and $Y$ be disjoint finite sets, $|X|=n$ and $|Y|=m$, so that we have the following bijections: $f:\mathbb{N}_n \rightarrow X$ and $g:\mathbb{N}_m \rightarrow Y$ I need to prove that ...
1
vote
4answers
52 views

Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
2
votes
3answers
34 views

Proof involving lcm and biconditional statement.

Suppose $a,b\in\mathbb{Z}$. Then $a = \operatorname{lcm}(a,b)$ if and only if $b\mid a$ Unsure of how to approach this problem.
2
votes
4answers
95 views

Prove $A \subset \emptyset \iff A = \emptyset$

How does one prove this? Can one prove by contradiction by saying: Let $A$ be any set such that $A$ contains at least one element. Now assume $A \subset \emptyset$. This is clearly absurd by the ...
0
votes
3answers
36 views

Proof for a non-conditional statement

I'm having a bit of trouble doing this proof. If $a\in\mathbb{Z}$, then $a^3 \equiv a \pmod 3$. I know how to do proofs if there were conditional statements but not sure how to prove this with ...
0
votes
2answers
71 views

If a converse of an implication is false, does this mean that the proof of that implication will always have an implication that is not reversible?

Let $f:X \rightarrow Y$ be a function and $B_1, B_2 \in \mathcal{P}(Y)$. Prove that $B_1 \subseteq B_2 \Rightarrow \overleftarrow{f}(B_1) \subseteq\overleftarrow{f}(B_2)$. My attempt: $\begin{align} ...
2
votes
2answers
32 views

Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $ I have some ...
-1
votes
1answer
27 views

Prove this statement (inequality)

$x,y,z$ $\in\mathbb R$ then $|x-y| \leq|x-z|+|y-z|$ Prove this statement. I thought it was the triangle inequality, but I can't seem to end up with the correct order.
1
vote
1answer
37 views

Correctly understanding truth table problem?

I'm typing up a solution set for an "intro to proof" course. One of the problems asks the student to "construct a truth table for $(P \implies Q) \implies (\neg P)$." I interpreted this as requesting ...
1
vote
3answers
34 views

Very simple proof help about integers

I was wondering if anybody could help me with proving something very simple: that $9n\ne6$ when n is any integer. It seems extremely intuitive but I don't know how to make it into part of a rigorous ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
1
vote
0answers
62 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
1
vote
2answers
42 views

Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$ (A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I. $$ I know that $A^{T}A$ = $I$, but where can I go ...
2
votes
2answers
174 views

Trying to prove that two angles are congruent in a isosceles trapezoid

I was given this assignment to do the following. Write a paragraph proof for the following scenario. Given: KLMN is an isosceles trapezoid. Prove: ∠LKM is congruent to ∠MNL The thing is that I ...
0
votes
2answers
45 views

Prove that $n<(3/2)^n$ for any $n$ with induction [closed]

need help with induction with inequality, I suck at it. $n<\left(\frac{3}{2}\right)^n$ for any $n$
2
votes
3answers
196 views

Wheel of Fortune Problem

The Summation formula is $$\sum_{i=1}^ni =\frac{n(n+1)}2$$ How is it that we know the integers $1,2,...36$ appear exactly $3$ times. And why do we multiply the sum by $3$ in the last part of the ...
2
votes
2answers
75 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
2answers
48 views

Proof on limit superior and limit inferior of a set

I understand the result intuitively but how can I prove this? For a given integral $n \ge 1$, let $A_n = \left\{\frac mn \mid m \in \mathbb Z\right\}$. Show that $\varlimsup_{n\to\infty} A_n = ...
0
votes
2answers
62 views

Proving Alternating Series Convergence

Suppose $x_n > 0$ and $\sum_{n=0}^\infty x_n$ is convergent. Prove that $\sum_{n=0}^\infty (-1)^nx_n$ is convergent. Any hints or starting points? So far I figured that I should show that the ...
1
vote
2answers
96 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty)$ be a nonnegative, ...
1
vote
2answers
57 views

Wondering if proof is proper

so I have been working on learning some new math in order to prepare for next year. I have been trying to learn proofs, and doing practice questions however the only problem is there are not answers. ...
5
votes
2answers
135 views

What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
4
votes
1answer
48 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
0
votes
3answers
30 views

Help with understanding definition of divisibility in this case.

I have a proof that shows that if $5 \mid xy$ then $5 \mid x$ or $5 \mid y$. It's pretty clear to me that I can just say that suppose $5 \mid x$, then $x=5a$, where $a$ is an integer. then $xy = ...
2
votes
0answers
18 views

Set with relative complement forms partition

Prove that if $S$ is a set and $ \emptyset \subsetneq A \subsetneq S $ then $\Pi = \{A , S-A \}$ is a partition of $S$. Proposed Solution: Since $ A \subsetneq S$ , we have $S - A \neq ...
0
votes
3answers
36 views

Help to prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$

Prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$ My attempt: $\begin{align} (x,y) \in (A \times B) \cup (C \times D) & \Rightarrow & (x,y) \in (A \times B) \vee ...
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votes
3answers
95 views

Examples of revisited proofs after new theorems are discovered… [closed]

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later? For instance, we know now that Pn+16< Pn+1 occurs infinitely often (where ...
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votes
1answer
27 views

Need help proving the statement

Assume that D ⊂ N and D ̸= ∅. Prove or disprove using a detailed structured proof, justifying every step: [∀x ∈ D, ∃y ∈ N, y < x] ⇔ [0 ̸∈ D] I have no idea how to prove a statement like that, I'm ...
2
votes
1answer
28 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
0
votes
1answer
52 views

Why are the different ways to write a universal statements equivalent?

Consider the following universal statements: $\forall a \in \mathbb{R}-\{0\}, a^2 > 0$ $\{a \in \mathbb{R} - \{0\}| a^2 > 0 \} = \mathbb{R}-\{0\}$ $a\in \mathbb{R}-\{0\} \Rightarrow a^2>0$ ...
2
votes
1answer
46 views

Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
1
vote
1answer
35 views

Decomposition of a function in positive and negative parts and its integrability

Is it true to say that $\int_\mathbb{R}|f(x)|dx<\infty\Rightarrow\int_\mathbb{R}f(x)=0$?