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
37 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
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1answer
27 views

Determining equivalence classes

I have done (a), pretty straight forward. I understand an equivalence class as all the elements in the domain that map to the same result in the co-domain. For example in (mod 3), [|0|] would be the ...
0
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1answer
29 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
1
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1answer
70 views

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$

Consider for real $a<b$ and real $A,B,C$, the function $f:[a,b] \to \mathbb R$ defined by $$f(x) = \begin{cases} A & x = a \\ B & x=b \\ C & a < x < b \end{cases}$$ I want to ...
1
vote
1answer
20 views

Writing skills: Proof of the relation between $\epsilon - \delta$ and open sets continuity

In order to check my math writing skills, I worked on writing the following basic proof. Theorem: If a function $f: X \to Y$ is continuous, then $G \subseteq Y$ is open implies that $f^{-1} (G)$ is ...
3
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2answers
48 views

Prove that the sequence $a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded.

Prove that the sequence $a_{0} = \frac{1}{2}, a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded. Assume that $0 < a_{n} < 1$ for every $n$ and $a_{0} = \frac{1}{2}$. Prof. used induction to prove that ...
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2answers
32 views

Proving Primness in a summation

I've been hitting my head against the wall for a little bit trying to figure out where to get started on proving (or disproving) the following: $\exists k \in \mathbb{Z} $ such that$ ...
3
votes
1answer
29 views

Is this exercise right, or something is wrong or missing.

I have to find the following limit For each positive integer $n$ define: $$a_n = \frac{1}{n}\left[\left(\frac{1}{n}\right)^2 + \left(\frac{2}{n}\right)^2 + ... + \left(\frac{n}{n}\right)^2 ...
0
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1answer
44 views

Verify if the following prove is right.

I need to prove the following: Suppose $f: [a,b] \to R$ is continous and $g: [c,d] \to [a,b]$ is differentiable. Define $F(x)= \int_{a}^{g(x)}f(t)dt$ for some $x \in [c,d]$. Prove that $F$ is ...
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2answers
16 views

Closed set contains the set and its closure proof check

The problem is as following: $A\subset X$. Show that IF $C$ is closed set of $X$ and $C$ contains $A$, then $C$ contains the closure of $A$ Here is my proof, but I dont know whether I have the right ...
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0answers
31 views

Prove this limit for a general $f(x)$ and $g(x)$

$f(x)$ and $g(x)$ have the following property: for all $\epsilon > 0$ and all $x$, $$ \text{if} \space 0 < |x - 2| < \sin^2(\epsilon^2/9) + \epsilon \space \text{then} \space |f(x) - 2| < ...
2
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3answers
31 views

Proving $f(n + 1) > f(n)$ and is f injective?

If I have a function $f:\mathbb N \to \mathbb N$ defined for every $n \in \mathbb N$ by: $$f(n) = (n+1)!-1$$ How would I prove that $f(n+1) > f(n)$ for every $n\in\mathbb{N}$? Would it be ...
3
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2answers
91 views

Is $x = 2$ is the only real solution for $a^x + b^x = c^x$ when $(a,b,c)$ is a pythagorean triplet?

Take any pythagorean triplet $(a,b,c)$, we know, by the definition that: $$a^2 + b^2 = c^2$$ But take $$a^x + b^x = c^x$$ Is $x=2$ the only possible solution $\in \Bbb R$ in this case? How can this ...
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3answers
39 views

Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$

$$A \cap B \subseteq A$$ My first step would be to write it as $(x \in A \land x \in B) \subseteq A$. Then I know by the following implication that is always true $P \land Q \implies P$. But I am ...
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vote
4answers
93 views

Proof: For every $x \in \mathbb R\,\exists n\in\mathbb Z,\,r\in[0,1):x=n+r.$

I still do not understand how to approach proofs. Any help would be appreciated. For every real number $x$, there exists an integer $n$ and a real number $r\in [0,1)$ such that $x = n + r$. ...
0
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0answers
37 views

Determine set of t-norms verifying a property

As recalled in Scholarpedia, a t-norm $\top(a,b)$ is a function from $\mathbb{R}^{2}$ to $[0,1]$ verifying the following properties: Commutativity: $\top(a,b)=\top(b,a)$ Associativity: ...
0
votes
1answer
56 views

Is $f : A \to P(A), a \mapsto \{a\}$ injective or surjective? [duplicate]

Given an arbitrary set $A$, let $f:A \to P(A)$ be the function defined for all $a \in A$ by "$f(a) = \{a\}$". How would you prove that $f$ is injective or surjective?
2
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1answer
37 views

Equivalence relation proof example. Starting off help

Now I need to prove its reflexive, symetric, and transitive! Now my biggest confusion is what do I let "a" equal? Obviously it will be an arbitrary element in N(sub 0). Any help would be great. ...
2
votes
1answer
45 views

having trouble saying this in a rigorous way

This is part of a problem I am working on and I have most of it figured out, but there is this one little piece that's kind of bothering me. I have a function $f:D \rightarrow R$ which is ...
1
vote
1answer
36 views

Proof involving Induction [duplicate]

Prove that for every integer n ≥ 1, we have $$ \sum_{i=1}^ni^3=\left(\frac{n(n+1)}2\right)^2 $$ Solve using Mathematical Induction, include the Inductive Step Base Case is that both the left and ...
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votes
5answers
82 views

Series and sequences convergence with a certain condition.

Let $\sum_{n=1}^\infty (a_{n})$ converge. Let $\{n_{k}\}$ be a subsequence of the sequence of positive integers. For each $k$ define $b_{k}=a_{n_{k-1}+1}+...+a_{n_{k}}$ where $n_{0}=0$. Prove that ...
0
votes
2answers
34 views

Proving that a certain series converge if and only if the a_n converges

I need to prove the following statement: Let $\{a_n\}$ be a sequence of real numbers.Prove that $\sum_{n=1}^\infty (a_{n}-a_{n+1})$ Converges iff $\{a_n\}$ converges. If $\sum_{n=1}^\infty ...
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2answers
54 views

Prove the convergence of a sequence involving integrals

I need to prove the following: Assume $f:[a,b] \to R$ is continous, $f(x)\leq0$ for all $x \in [a,b]$, and $M=sup\{f(x):x \in [a,b]\}$.Show that: $$\{[\int [f(x)]^{n}dx]^{1/n}\} \to M$$ This result ...
0
votes
1answer
25 views

“Natural” Homeomorphisms, Retracts and Knots

I have been trying to prove the following result for a few days now, and have made some amount of progress, but now I'm struggling. This is what I'd like to prove: Every proper knot, K has a retract ...
1
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3answers
78 views

Is this correct limit proof for $\sqrt{x}$

Prove: $\displaystyle \lim_{x\to 1} \sqrt{x} = 1$ $\displaystyle |\sqrt{x} - 1| < \epsilon \space \text{such that}\space |x - 1| < \delta$ Let $|x - 1| < 1 \implies |x| < 2 \implies -2 ...
0
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0answers
19 views

Prove equal cardinalities

So I asked a question a while ago here. Help on the Inclusion Exclusion principle and explaining cardinality And I understand the solution. However, I am rather perplexed. I thought the way to ...
1
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2answers
36 views

Positive convergence of terms proof

Let $(s_n)$ be a sequence of positive terms such that the sequences of ratios $(\frac{s_n+1}{s_n})$ converges to $L$. Prove that if $L>1$, then $\lim s_n=+\infty \!\,$ So I know I have to use the ...
1
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3answers
38 views

Mathematical Induction and Proof

How would one prove this using mathematical induction. Suppose that $i$ is a number such that $i^2 = −1$. Prove that for any integer $n ≥ 0$, we have $$ (\cos(x)+i\sin(x))^n =\cos(nx)+i\sin(nx). $$
0
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1answer
46 views

Subset proper subset proof help in discrete math

I am very lost on this question for my discrete math class: Let A and B be sets. Suppose A contains at least 2 elements. Prove that if every proper subset of A is a subset of B, then A is a subset of ...
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0answers
33 views

proofs without using symbols/notations

I am looking for a big list of mathematical proofs which only use pure text, without introducing symbols and extra notations.
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2answers
26 views

Need help proving a surjection / uncountability

Here is what I think. A) To prove a surjection.. it goes like this. Take an arbitrary b in the Real Numbers (codomain). Let a = "SOMETHING" and we want to show f(a) = b. Now since the function is ...
0
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4answers
49 views

For any $ n \in \mathbb N$, there is an odd integer m such that $n^2 < m < (n+1)^2.$

Thanks for the help for the previous proof. Now I am stuck on this statement I need to prove. For any natural number $n$ there is an odd integer $m$ such that $n^2 < m < (n+1)^2$. I got ...
0
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1answer
26 views

Cardinality problem with multiple sets

What we are given in part A) This is part B) What we are asked to prove My best idea is that if you evaluate what the proof is, it turns into what is given except A is Ai and B is Ai+1 and all ...
1
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6answers
74 views

Prove: For any integer $n \geq 2$, there is an odd number P such that $2n \lt P \lt 3n$

I am in high school and had this for a homework problem. I got it wrong, but the teacher did not post the correct answer. Any help would be appreciated. It is about writing proofs. Prove that for ...
0
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2answers
35 views

Cardinality with Cartesian Cross Product problem

A and B are finite sets Prove that |AxB| = |A||B|. I need a solution/hint. I suspect that the answer has to do with the fact that the we can say that |A| = |B| and then from that say = |AxB|. I ...
1
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0answers
26 views

How to apply Fubini's theorem in proof of Osgood's lemma

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots ...
1
vote
2answers
86 views

Prove by induction the predicate (All n, n >= 1, any tree with n vertices has (n-1) edges).

I'm stuck on this problem, posting my progress so far below. I've looked at similar questions here and here, but neither seem to directly prove the predicate by induction, with a base case followed by ...
0
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2answers
41 views

How to formally state and prove vacuous truth?

How to show in a proof that a statement is vacuously true because "if $\alpha$ then $\beta$", and also prove $\alpha$ is false, in a formal way? and also particularly, how to structure such proofs? ...
0
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1answer
36 views

Formating equations in a scientific paper

When I write derivations that spread over several lines, I use the following format $V(x) = x^2 - 2x + 1$, from (3) $= (x-1)^2$ $> 0, because x > 0$ ...
0
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2answers
26 views

Help on the Inclusion Exclusion principle and explaining cardinality

I want to prove the inclusion exclusion principle: |A∪B|=|A|+|B|−|A∩B| where A and B are finite sets. However I'm confused about one thing. I've learned that two cardinalities are equal if there is a ...
2
votes
2answers
45 views

Mathematical Induction Proof - Exponent with n in denominator

Use mathematical induction to prove the following: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{n}{2^n}=2-\frac{n+2}{2^n}; n ∈ N $$ I am having trouble figuring out how to solve this with an ...
30
votes
2answers
559 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
0
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2answers
20 views

Proving a surjection. Clarification

I just want to make sure this is all correct. So my definition of a function $f:A\to B$ being a surjection is: For all $b \in B$, there exists an $a \in A$ such that $f(a) = b$. Now the ...
2
votes
1answer
29 views

Prove that stabilizer subgroups of G are conjugate to each other

Suppose that a group $G$ acts on a set $X$. Show that if $x_1$ and $x_2$ in X are in the same $G$-orbit, then their stabilizer subgroups of $G$ are conjugate to each other. My proof: Assume ...
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0answers
32 views

Proof of predicate logic identity using quantifiers

I am attempting to prove this identity using only basic predicate logic rules. $\left ( \forall x A \right )\rightarrow B = \exists x \left ( A\rightarrow B \right )$ I understand that in order to ...
0
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0answers
39 views

Greatest common divisor / euclidean algorithm linear combination proof

Consider integers $m$ and $n$, not both 0. Show that gcd$(m,n)$ is the smallest positive integer that can be written as $am + bn$ for integers $a$ and $b$. I'm confused on what exactly to do--I'm ...
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3answers
52 views

Prove that in a ring with at least two elements $0\neq 1$. [closed]

Let R be a non-trivial ring then prove $0\neq 1$.
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0answers
62 views

How to use a very complicated theorem for proving simpler statements without falling into a loop?

There are some too complicated theorems in mathematics which have very complicated proofs in hundreds of pages. There are few mathematicians who are aware of the entire proof of such theorems in full ...
4
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0answers
38 views

Another proof question for real analysis

Let $a, b, c \in \mathbb{R}$. Prove if $a + b = a$ then $b = 0$. Suppose that $a + b = a$. Then $a + b - a = a - a = 0 = b$ by the inverses law for addition. By the Identity law for addition it ...
0
votes
2answers
37 views

Recursive definition proof

I'm having trouble proving the following: $a_0 = a_1 = 1$ and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 2$. Prove that all the terms $a_n$ are odd integers. It makes sense since an odd number is of the ...