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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21 views

Antiderivative of an even function

I'm faced with an issue in terms of antiderivatives of even and odd functions. Define $f \in C[-a,a]$ where $a>0$. Let $f$ be an even function on $[-a,a]$. We wish to show that $$\int_{-a}^a ...
3
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1answer
49 views

Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$

It seems obvious that $$\forall n \in \Bbb Z^+, \sum_{i=1}^n{1} = n $$ However, I'm having trouble coming up with a formal proof for this. Given a concrete number like $4$, we can say that ...
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4answers
51 views

Show that if $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$.

Show that if $m$ and $n$ are integers such that $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$. I am not sure where to begin.
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2answers
27 views

prove that 2 does not go into $n^2 – 2$ without a remainder for odd $n$.

Prove that $2$ does not go into $n^2 – 2$ without a remainder for odd $n$. How do I approach this?
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1answer
19 views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
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1answer
22 views

Proving the existence of a Bijection between Cartesian Products of Sets by Induction

Prove by induction that for any sets $A_1, \ldots , A_n$, there is a bijection from $(((A_1 \times A_2) \times A_3) \times \ldots \times A_n)$ to $A_1 \times (A_2 \times ( \ldots (A_{n-1} \times A_n) ...
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2answers
36 views

Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
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1answer
56 views

What is the contraposive of this statement?

I have to prove the negation of this statement: $$\forall a,b,c\in\mathbb{Z}{\;if\;a\;|\;b\} $$ But the fact that there is a "and" is very disturbing. I think that I am missing something because my ...
0
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1answer
34 views

how to prove $pr_i(\alpha \setminus \beta) \supseteq pr_i\alpha \setminus pr_i\beta$

For those who are not familiar with the syntax $pr_i \alpha = \{ pr_i(a,b) / a \alpha b \} \text{ for }\alpha \subseteq A \times B$ which is same as $\begin{cases} (x= pr_1 \alpha) \Leftrightarrow ...
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0answers
29 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
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4answers
49 views

Prove that a continuous real function with finite limits is bounded

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Assume that $\lim_{x \rightarrow \pm \infty} f(x)$ exist and are finite. Prove that $f$ is bounded. So to show that $f$ is bounded, I ...
2
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1answer
39 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
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0answers
15 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
13
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3answers
323 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
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1answer
26 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
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0answers
44 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
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0answers
57 views

On correctness of induction proof

I want to prove a certain property $\mathsf{P}$ on every multiaffine polynomial in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$. Supposing I show property $\mathsf{P}$ to be valid at $n\geq9$ variable ...
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0answers
30 views

Proof: A+B is upper triangular [on hold]

Assume A and B are nxn matrices. Prove that A+B is upper triangular.
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2answers
36 views

Is $\sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous?

$f(x) = \sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous because: Proof: $f(x)$ is a continuous function on $[0, 1]$, which is a closed interval, so $f$ is uniformly continuous on ...
4
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0answers
24 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
1
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1answer
35 views

Clarification on Cantor Diagonalization argument?

My book is Discrete Mathematics and its Applications. This is its section on Cantor's Diagonalization argument I understand the beginning of the method. The author is using a proof by ...
0
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1answer
20 views

How to prove the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?

Let $e_1, e_2, e_3$ be an orthonormal basis. How to prove that for any vector $u$, $$u = (u, e_1)e_1 + (u, e_2)e_2 + (u, e_3)e_3,$$ i.e., the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?
2
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3answers
56 views

How to approach this proof problem, what proof to use, what assumption to use?

This is a problem from Discrete Mathematics and its Applications Here is the definition of rational that my book uses Usually when I approach this type of a problem, I can find a type of proof to ...
0
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0answers
22 views

Would it be necessary to have another proof within the proof by cases in this problem?

This is a problem from Discrete Mathematics and its Applications I am using Proof by Cases. This is my book's definition on it. Here is my work so far I tried to leverage without of generality ...
1
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1answer
38 views

Induction on the size of the set?

Show that every non-empty finite set of real numbers has a maximum. (Hint: induction on the size of set). I'm not exactly sure how to approach this. I'm familiar with induction, but I don't know ...
0
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1answer
24 views

Proof of Z-transform of n

How can I prove the following Z-transform: $$ Z\{n\} = \frac {z} {(z-1)^2} $$ As a tip, I was told to use the 'Multiplication in time'-property of the Z transform, which is the following: $$ ...
0
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2answers
29 views

Next step to take in this proof by contradiction?

This is a problem from Discrete Mathematics and its Applications Here is my work so far It's similar to this other question I had Next step to take to reach the contradiction?. I am assuming ...
2
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3answers
47 views

Can someone verify my direct proof that if A is a subset of B, AU B = B?

This is a problem from Discrete Mathematics and its Applications I am trying to use a direct proof to do this problem. Here is my book's explanation/section on direct proof Here is my work so ...
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0answers
11 views

Subsets being cones

I am trying to self-study convex optimization and still trying to get into the gist of it. There is a question in my text as follows: Let $V$ be the set of sequences whose terms are contained in ...
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4answers
135 views

Proving that $p_1p_2\mid n$ iff $p_1\mid n$ and $ p_2\mid n.$

Let $p_1$, $p_2$ be distinct primes. Using the Fundamental Theorem of Arithmetic prove that a natural number $n$ is divisible by $p_1p_2$ if and only if $n$ is divisible by $p_1$ and $n$ is divisible ...
1
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1answer
40 views

Question about maps in partially ordered sets

I am struggling with a proof related to the mapping of posets. Let $P_1$and $P_2$ are posets, and $f$ an order preserving map from $P_1$ to $P_2$. $$g(Z):= f^{-1}[Z]$$ $g$ goes from the set of all ...
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0answers
30 views

Verify my proof: if $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive.

Could someone verify my proof and my writing? Proposition: If $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive. Definition 1: A binary ...
0
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1answer
7 views

question about vacuous truth and function

I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of ...
1
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7answers
50 views

Induction proof of $1 + 6 + 11 +\cdots + (5n-4)=n(5n-3)/2$

I need help getting started with this proof. Prove using mathematical induction. $$ 1 + 6 + 11 + \cdots + (5n-4)=n(5n-3)/2 $$ $$ n=1,2,3,... $$ I know for my basis step I need to set $n=1$ but I ...
0
votes
2answers
44 views

If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$

Suppose that $k$, $n$, and $d$ are integers and $d$ is not $0$. Prove: If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$. You may not use the theorem stating the following: Let ...
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1answer
40 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...
2
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0answers
19 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
2
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0answers
66 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
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3answers
81 views

Real Analysis Proofs

I am taking a Real Analysis class using the textbook Analysis with an Introduction to Proofs, $5^{th}$ Ed. by Steven Lay. So far I am not understanding the proofs at all. Does anyone know of any good ...
2
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2answers
75 views

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction [duplicate]

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction My proof so far: Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ Base Case $P(1):$ LHS = $1^3 = 1$ ...
2
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0answers
53 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
0
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1answer
48 views

Prove that $nCr = n(n-1)(n-2)\cdots(n-r+1)/ 1\cdot2\cdot3 \cdots r$ is an integer for all positive integral $n$ and for all integers $r \geq 0$.

Prove that $nCr =\frac{ n(n-1)(n-2)\cdots(n-r+1)}{ 1\cdot2\cdot3 \cdots r}$, is an integer for all positive integral values of $n$ and for all integers $r \geq 0$. Can someone please explain it to ...
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2answers
62 views

Inequality $\prod\limits_{r=1}^{- \infty}(1+(\frac{1}{2})^r)<\frac 52$ [closed]

Prove this inequality. $\prod\limits_{r=1}^{- \infty}\left(1+\left(\frac{1}{2}\right)^r\right)<\dfrac 52$ I have tried to prove it using induction but it is not coming.
2
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1answer
78 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
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2answers
60 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
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2answers
31 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
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1answer
55 views

Attempt to proof the Cantor-Bernstein theorem

I've found a proof of the Cantor-Bernstein theorem in Kleene's 'Introduction to Metamathematics' (1952) in §4 Thm A. I must admit I don't understand its essence but I was wondering if the proof could ...
1
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1answer
24 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
1
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1answer
30 views

How to display one to one correspondence?

This is a problem from Discrete Mathematics and its Applications Here is the book's definition of countable/not countable For 2a, I came up with the fact that the set is countably infinite. What ...