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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0answers
30 views

Determining if two bounds are true

Question says assume $f$ and $g$ have a domain of the integers, and target space of the real numbers. $f$ and $g$ are bounded. Prove if the following statements are true or give a counterexample: if ...
1
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1answer
24 views

Can I use logical equivalence instead of biconditional in proofs?

My textbook defines the symbol <=> to mean equivalent to, has the same solutions as or if and only if. It defines the symbols => and <= to mean implies or leads to. The textbook does not use the ...
1
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1answer
39 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
3
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2answers
52 views

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $. For this question, I think that I would use the completeness axiom. As $3$ is greater than $2$, so $S$ has a ...
-1
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1answer
83 views

f and g are bounded with domain of integers and target the real numbers . If f/g is bounded, then g/f is bounded.

I have come up with two bounded functions f = 1/x^2+1 and g = 1/x^2+2 and these tell me that g/f is also bounded. However, I am having trouble writing a proof or proving that g/f is not bounded by ...
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1answer
86 views

f and g are bounded . if 1/g is bounded, then f/g is bounded.

I would like some help understanding how to go about this question. I think that f/g is not bounded, but I cannot figure how to show that f/g is not bounded.
3
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3answers
85 views

Prove $\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$ using $\delta-\epsilon$ definition of limit

I need to prove that the $$\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$$ using $\delta-\epsilon$ definition of limit. Now, I have started with a discussion, saying that what we want is that if $\left| x - ...
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4answers
624 views

How to learn/speak “mathematical english”?

Good day! I was wondering if there is a good way to learn "maths in english". I am studying mathematics in Germany (I am from Germany, so english is not my native language) and have recently started ...
2
votes
2answers
189 views

Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$

Prove that the sequence $s_n = \frac{1}{n}$ converges to $0$. I am writing this proof in order to help other people to understand better how to prove if a sequence converges and in particular why ...
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2answers
50 views

Book Recommendations for Writing Proofs

As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an ...
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1answer
45 views

Midpoint of a set as Mean? [closed]

Given a set of an odd number of terms: $x = \{a, b, c, ..., \}$ Consisting of $n$ elements. How is the midpoint of the set. A proof and explanation would be helpful: $$\frac{a + b + c + ... }{n} ...
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3answers
64 views

Isn't $\mathbb{P}$ already a probability measure, so what is there to prove?

Follow-up to Probability measure over finite sample space. This is a theorem from Casella and Berger's Statistical Inference: Let $S = \{s_1, \dots, s_n\}$ (sample space) be finite and $p_1, ...
1
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1answer
30 views

Proving a sequence is Cauchy (and convergent) by an infinite geometric sequence (something also with Lipschitz)

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics so Lipschitz hasn't passed the course yet. I should be able to prove this ...
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2answers
31 views

Prove Alternating Series Approximation

Prove if $S=\sum_{n=1}^{\infty}a_{n}$ is an alternating series with $\left | a_{n+1}\right | < \left | a_{n} \right |$, and $\lim_{n\to\infty}a_{n}=0$, then $\left |S-(a_{1}+a_{2}+\cdots+a_{n}) ...
0
votes
1answer
30 views

How to proove Hammer Split-graph Theorem?

Let $G=(V,E)$ be a Split Graph with $|V| \geq 4$. Then how to prove that: No induced sub-graph of G with 4 Vertices is a cycle with length 4 OR a pair of not incident edges? Well it must be from ...
2
votes
1answer
50 views

What are the requirements for a statement to have a constructive proof?

In general when trying to solve an excersise, or construct a proof, I always find myself looking at what strategy should I take to complete the proof. Many times I try to solve the excercise with a ...
0
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1answer
23 views

Prove that a tournament is irreducible if and only if it is strongly connected

If a graph is irreducible, by definition there will be no source or sink and it will be strongly connected. Is my proof above good and how do I prove the converse?
3
votes
1answer
57 views

No truth function that is expressed by a formula that uses only implication and equivalence connectives

I proved the following statement by induction: Let $A$ be a propositional formula which uses only the connectives $→$ and $↔$. Prove (by induction on the complexity of $A$) that if every ...
1
vote
1answer
36 views

If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...
2
votes
1answer
37 views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
1
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1answer
68 views

Proving by induction on the length of a propositional formula?

I'm having a little trouble understanding the following proof question because I'm unsure what defines the 'length' of a propositional formula, I've seen multiple definitions whether it's the number ...
1
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2answers
55 views

How to prove the following inequality: $\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$

As a part of my practice for an upcoming mid-term, I managed to simplify the following inequality to what you see here: $$\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$$ And honestly I'm ...
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2answers
35 views

How to formally prove that an element belongs to a sequence of sets.

Take any $\delta \in [ \frac{1}{2}, 1)$, I want to show that there always exists an $n$ s.t. $\delta \in [\frac{1}{2}, 1 - \frac{1}{n}) $. Can one obtain an explicit relationship between $\delta$ and ...
4
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0answers
71 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
2
votes
0answers
37 views

Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...
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2answers
25 views

Determine the rank of the linear map given by a $n \times n$ matrix , dependend on n. Proof by induction

The task: Let $$ A:= \begin{pmatrix} 1 & a & a & ... & a\\ a & 1 & a & ... &a \\ a & a & ... & a & a\\ ... & ... &... & 1 & a \\ a ...
2
votes
1answer
57 views

Induction implies by well-ordering

A problem in Spivak's Calculus, ch 2-10, asks to prove induction by the well-ordered principle. I have read a number of answers to that question on this site, but I would like to see the proof in a ...
4
votes
1answer
40 views

The set of algebraic numbers is countable: is this proof correct and well written?

Problem: prove that the set of all algebraic numbers is countable. My proof: Let $f: \bigcup^{\infty}_{n=1} \mathbb{Z}^n \rightarrow \mathcal P(\mathbb{C})$ be a function associating an ordered ...
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1answer
59 views

Did I solve exercise 4.5.4 (b) of 'How to Prove it' by velleman correctly and concisely?

4.5.4 Suppose R is a strict partial order on A. Let S be the reflexive closure of R. (b) Show that if R is a strict total order, then S is a total order. Suppose R is a strict total order. ...
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7answers
117 views

Prove $ \{(p \lor q) \land (p \implies r) \land (q \implies r) \} \implies r$ is a tautology using logical properties

I spent quite a bit of time on this and have little to no ideas on how to proceed. Using the conditional laws and De Morgan's law, I got to $$( \sim p \land \sim q) \lor (p \land \sim r) \lor(q ...
2
votes
1answer
38 views

what is wrong with this proof? (proving the transitive property of Big O)

So the problem is if $f(n) \in O(g(n))$,and $g(n) \in O(h(n))$ then $f(n) \in O(h(n))$ Assume $f(n) \geq 0, g(n) \geq 0, h(n) \geq 0$ Proof: From assumptions, ...
0
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1answer
58 views

Show that A is an open subset of M

If $m\in\mathbb{N}$, $M=\{0,1\}^{\mathbb{N}}$ and $A \subseteq M$ is an open set of sequence where the number 1 appears at least $m$ times. Show that $A$ is an open subset of $M$. I wanted to show ...
2
votes
1answer
45 views

Complex analysis, residues of function

If $f(z)$ has residue $b_1$ at $z=z_0$, show by example that $[f(z)]^2$ need not to have residue $b_1^2$ at $z=z_0$ What I tried Suppose that $f$ is analytics in the neighborhood of $z_0$ and ...
1
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1answer
76 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) ...
1
vote
1answer
33 views

On existence of square root of positive elements of a unital $C^*$-algebra

Given a unital $\mathcal{C}^*$-algebra $A$ and a positive element $a \in A$, I am trying to prove the existence of a square root $a^{\frac{1}{2}}$ i.e. a positive element $b \in A$ such that $b^2 = ...
0
votes
1answer
50 views

Prove that length of a curve is finite / infinite

I have problems proving the following: Let $\alpha > 0 $. Consider the curve $\gamma : [0,1] \to \mathbb{R}^2$ given by $\gamma (0)=(0,0) , \gamma(t) = (t^\alpha \cos ( \frac {1}{t}), t^\alpha ...
1
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1answer
30 views

Converging subsequences and subsets having infinite elements

We have a metric space (V,d). Proof that the following two properties are equivalent. a) Every sequence $a_n \in V$ has a subsequence which converges to a element $x \in V$ b) For every ...
0
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1answer
36 views

Proving that a solution exists

Proof that there exists a $x>0$ with $x \in \mathbb{R}$ s.t. $\sin(x) = \frac{x}{2}$ I tried to use the intermediate value theorem, but I don't know how to apply it correctly. Obviously ...
0
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4answers
48 views

How can I prove that this matrices statement is false?

How can I prove that this is not true: If for matrices A, B and C, AB=AC and A is not the zeroth matrix, then B=C.
3
votes
1answer
27 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
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2answers
40 views

Prove using Hilbert calculus $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$, formal proof.

Prove: $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$ Using Hilbert Calculus Format of solution: Step (my understanding) Solution: (1) $\forall x(Px\rightarrow x\equiv a)\vdash ...
3
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2answers
71 views

Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
2
votes
4answers
40 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
2
votes
1answer
40 views

Using lipschitz estimate to show $|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$

Assume $(f_n)$ is a sequence of functions that are continuous on $[a,b]$ and differentiable on $(a,b)$. Then using Lipschitz estimate to prove that $$|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq ...
5
votes
1answer
33 views

Proof check, showing pointwise convergence

My problem is this: For $x \in [0,\frac{\pi}{2}]$, $f_n(x) = \frac{nx}{1 + n\sin(x)}$ Find the pointwise limit of $(f_n)$ for all $x \in [0, \frac{\pi}{2}]$ I am not sure if the way I constructed ...
0
votes
0answers
15 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
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2answers
39 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
2
votes
1answer
25 views

Prove that if $f$ is an invertible function and $g$ is an inverse, then the codomain of $g$ is equal to the domain of $f$ and vice versa

I am trying to show, without using the bijection properties, what is above. Assume $f$ is an invertible function and $g$ is an inverse of $f$. For $f \circ g $ to be well defined then the image of ...
0
votes
1answer
21 views

Placement of quantifiers in a symbolic statement

I have the statement: Let $A_n$ be an indexed set of numbers defined by $A_n = n\mathbb{Z}_{\geq m}$ for $n,m \in \mathbb{Z}$. Consider the claim C: For all $n$, if $x,y$ is in $A_n$, then ...
1
vote
2answers
45 views

Proving the well ordering principle

THe well ordering principle has that every subset of $\mathbb{Z}^+_0$ has a least element. or if $S$ is a non-empty subset of $\mathbb{Z}^+_0$ and $S = \{a_1, a_2, a_3 ... a_n\}$, then there is a ...