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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2
votes
2answers
37 views

Proving that $f$ is a bijection.

Here is the question: Suppose $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ satisfy $$\forall x\in X.(g\circ f)(x)=x,\,\forall y\in Y.(f\circ g)(y)=y$$ Prove that $f$ is a bijection, with ...
1
vote
0answers
26 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
2
votes
1answer
54 views

Question about $\frac {\Gamma'(z+1)}{\Gamma(z+1)}$

If $\psi (z)= \log\Gamma(z+1)$ Prove that : $$\psi(n)+\gamma=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ My Proof : $$\psi (z)= \frac {\Gamma'(z+1)}{\Gamma(z+1)}=-\frac{1}{z+1}-\gamma + \sum_{n=1}^\infty ...
0
votes
2answers
29 views

Prove that a certain sequence of partial sums (involving integrals) converge.

I have to prove the following: Define $\gamma_{n}= 1+1/2+1/3+...+1/n-\int_{1}^{n}\frac{1}{t}dt$.Prove that $\{\gamma_{n}\}$ converge. I need your help because I don't know how to involve the algebra ...
8
votes
2answers
99 views

Let $G$ be a group, and $H$ a subgroup of $G$. Let $a, b \in G$. Prove $Ha=Hb$ iff $ab^{-1} \in H$.

Let $G$ be a group, and $H$ a subgroup of $G$. Let $a, b \in G$. Prove $Ha=Hb$ iff $ab^{-1} \in H$. $\rightarrow$ If $Ha=Hb$, then $h_1a=h_2b$ for some $h_1, h_2 \in H$. So, $ab^{-1} = ...
0
votes
4answers
109 views

Proof that $ 3 > (1+\frac{1}{n})^n \geq 2$

I am studying computer science in first term, and i got a task that i was not able to solve for a long time now. I have to prove that $ 3 > (1+\frac{1}{n})^n>=2$ for every $n \in ...
1
vote
5answers
109 views

Writing clear proofs involving multiple theorems and conditions

Suppose the problem is that given $A$ and $C$ holds, prove $D$ holds. Some theorems that we can use are: $A \to B$ $(B,C) \to D$ I feel what I said may be unclear: Because $A$ holds ...
1
vote
1answer
45 views

Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
2
votes
3answers
91 views

Validity of this proof that any continuous function with domain and range in [0,1] must have a fixed point.

The following proof was given in a solutions manual to a question asking to prove that a continuous function with domain and range in $[0,1]$ must have a fixed point: Consider the function $F(x) = ...
0
votes
3answers
32 views

The subspace $S ⊆ \mathbb{R}^n$ has linearly independent vectors $u_1,…u_k$. Show that any basis for $S$ must have at least $k$ vectors.

Let $S ⊆ \mathbb{R}^n$ be a subspace. Say that $u_1,.....u_k ∈ S$ are linearly independent vectors. Show that any basis for S must have at least k vectors. "Say that $u_1,.....u_k ∈ S$ are linearly ...
1
vote
0answers
33 views

How to prove that a function f(x) is O(g(x)), using the definition (finding C and k)

We say that $f(x)$ is $O(g(x))$ if $$(∃C ∈ \mathbb(R)❘)(∃k ∈ \mathbb(R)❘)(∀x ∈ \mathbb(R)❘)$$ $$(x ≥ k → |f(x)| ≤ C · |g(x)|)$$ In English: We can find $C$ and $k$ so that, once we get past the “small ...
0
votes
1answer
27 views

Need help on a example about proof on functions and sets

I need some help to prove the problem below: Suppose $g$ is a function from $X$ to $Y$ and $f$ is a function from $Y$ to $Z$. $A$ and $B$ are subsets of $X$. Prove that if $A$ is a subset of $B$ then ...
1
vote
2answers
99 views

Prove: Use the triangle inequality to prove that for all $x, y, | |x| − |y| | ≤ |x − y|$ [duplicate]

Prove: Use the triangle inequality to prove that for all $x, y, | |x| − |y| | ≤ |x − y|$ Proof: If $x ≥ 0$ and $y ≥ 0$, then both sides of the inequality are the same. Also if $x ≤ 0$ ...
0
votes
1answer
115 views

Define a relation ~ on ℕ by a~b if ab is a perfect square

So, for this problem: a. Prove that ~ is an equivalence relation on ℝ². (I'm not sure if this is a typo on my professor's part since we are defining a relation on ℕ.) b. Describe the equivalence ...
1
vote
3answers
53 views

Supremum of the product of sets

Let $A, B$ be subsets of positive real numbers that are bounded above, and let $A\cdot B=\{ a b : a\in A, b\in B\}$. Show that $$ \sup (A\cdot B) = \sup A \sup B. $$ Proof: This is obvious. It is ...
1
vote
1answer
54 views

Define a relation ~ on ℝ² by (x,y)~(w,z) if x+y=w+z

So, it comes in two parts: a. Prove that ~ is an equivalence relation on ℝ². b. Give a geometric description of the partition of ℝ² formed by the equivalence classes. For a, I have to prove that ~ ...
0
votes
1answer
45 views

The properties of relations $xRy$ if $x\ge y^2$ and $aRb$ if $a|b$.

I am given these 5 questions about the relations $xRy$ if $x\ge y^2$ (on real numbers), and $aRb$ if $a|b$ (on $\mathbb N$). a. Find the domain and range of R. b. Prove or disprove that R is ...
1
vote
0answers
35 views

Suppose A and B are sets. If $|A| = m$ and $|B| = n$, then how many relations are there from $A$ to $B$?

Coudn't I say that it is $m\times n$ relations because the collection of all ordered pairs is the Cartesian Product $A\times B$?
0
votes
0answers
39 views

Measure Theory: Basic Proof

Recently I've been trying to write better as a mathematician and now details that seemed to be apparent before were just so because I was dismissing them. I would like to know if the following proof ...
0
votes
2answers
61 views

An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$

Problem: Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon ...
0
votes
2answers
54 views

Proof of Continuity of $f(x) = x^2 + x - 1$

Am supposed to show proof that $f(x) = x^2 + x - 1$ is continuous for all real numbers a. Doing the general Delta-Epsilon proof, I have and understand how to prove $x^2$ alone. In particular I am ...
2
votes
3answers
90 views

Proving $\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$ by induction

Prove that $$\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$$ for all $n\in \mathbb{N}$ where $n\geq2$. I've already proven the base case for $n=2$, but I don't know how to make the next step. Is the ...
1
vote
1answer
41 views

How to prove that $\forall n\in \mathbb{N}$, $\sum ^{n}_{i=1}i^{3}=\frac {n^{2}(n+1)^{2}}{4}$? [duplicate]

Use mathematical induction to prove that $\forall n\in \mathbb{N}$, $$\sum ^{n}_{i=1}i^{3}=\dfrac {n^{2}(n+1)^{2}}{4}$$ $$\begin{align*} \sum_{k=1}^{n+1} k^3 &= \sum_{k=1}^{n} k^3 + (n+1)^2 ...
-1
votes
1answer
57 views

How to prove using math induction that $\forall n\in \mathbb{N}$, $\sum ^{n}_{i=1}i^{2}=\frac{1}{6}n\left( n+1\right) \left(2n +1\right)$? [duplicate]

Use mathematical induction to prove that $\forall n\in \mathbb{N}$, $$\sum ^{n}_{i=1}i^{2}=\dfrac {n\left( n+1\right) \left(2n +1\right) }{6}$$
1
vote
1answer
28 views

Completeness implies an infimum

Let $F$ be an ordered field where every non-empty subset which is bounded above has a supremum. Prove that every non-empty subset which is bounded below has an infimum. Let $S_1 = \{-x \mid x \in ...
3
votes
2answers
54 views

Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$

How to deal with $m+\frac{4}{m^2}\geq3$ for every $m > 0$ ? I multiplied both sides by $m^2$ and got $m^3+4-3m^2\geq0$ and have no idea how to continue with this
0
votes
1answer
29 views

Demonstration of strong induction using ladder rungs

My textbook illustrates strong induction using a ladder analogy as follows: Suppose we can reach the first and second rungs of an infinite ladder, and we know that if we can reach a rung, we can reach ...
2
votes
3answers
67 views

$A$ is dense in $[0,1]$ and $f(x)=0$ for all $x$ in $A$ then the integral is zero

I need to prove the following: Let $A$ be a dense set in $[0,1]$.Suppose $f:[0,1] \to \mathbb{R}$ is Riemann integrable and $f(x)=0$ for all $x \in A$. Show that $\int_{0}^{1} f(x) dx=0$ My ...
3
votes
4answers
45 views

How to prove that $m+(4/m^2)>=0$ for every m greater or equal to 0 [duplicate]

I have a problem with proving this simple theorem. I've already figured out that the best strategy is to factorise is, so that to get eg. a square or another expression that from the definition has to ...
3
votes
3answers
54 views

Proof that an odd integer multiplied by 3 and squared is always odd

I'm working with a proof in a discrete structures CS course, and I am a little confused by how to build up some logic for the argument. Currently we're working with symbolic logic, the problem ...
0
votes
3answers
58 views

How do I write a variable, x, when I mean 'any x' so that it's clear I don't mean a particular number.

In high-school, we usually used letters (literals?), such as $y$ to designate particular unknown numbers. In functions, $y$ could designate various numbers, but it seems to me that in these cases $y$ ...
2
votes
2answers
334 views

circular reasoning in proving $\frac{\sin x}x\to1,x\to0$

The classic proof for $\frac{\sin x}x\to1,x\to0$ is using a squeezing theorem based on arguments about areas of circles. But as far as I know, all deduction of formula of circles' area is based on ...
0
votes
1answer
53 views

Proof of the number of the leaves in a full binary tree

I need to proof by induction that at full binary tree there are $\frac{n+1}{2}$ leafs if $|V|=n$. So, I won't write you the whole proof, just my idea, and I'd like to know if this OK... So we ...
3
votes
2answers
67 views

Integer solutions of the equation $7(a^2+b^2)=(c^2+d^2)$

What are all the integer solutions of the equation $7(a^2+b^2)=(c^2+d^2)$ First thing to note is that $c=7C$ and $d=7D$ and substituting it in the original equation yields an equation that is ...
1
vote
3answers
44 views

If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. Assume, x, y ∈ ℝ # Domain assumption ...
1
vote
0answers
18 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
0
votes
0answers
17 views

The Answer to the problem Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N [duplicate]

I need to Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N My attempt: We first define $B=\{d>0: divisor of N\}$, ...
0
votes
0answers
19 views

help with proof involving matrix derivations

So, Ive been trying to learn the research in a particular article, which can be read http://www.sciencedirect.com/science/article/pii/0024379580902219# Specifically lemma 2. So far, I have understood ...
2
votes
1answer
49 views

Linear Algebra: Symmetric matrices, diagonalization (help with proof)

I need a bit of help with an IFF proof, here it is: {Let X be a symmetric n × n-matrix. Show: $$X=Y^2$$ for some symmetric matrix Y iff X has only non-negative eigenvalues. } My thinking: This ...
0
votes
1answer
32 views

How to prove that path in directed tree is directed path?

So I have a directed tree where I have a path that begins in the root of tree and leads to any vertex. I have to prove that this path is a directed path.
0
votes
1answer
50 views

How to prove that graph has cycle?

Let $(V,E)$ be a graph where between each two vertices $v_1,v_2\in V$ there exists only one path. Then The graph has no cycles. Adding a new edge creates a cycle. I have no idea how it could be ...
1
vote
2answers
57 views

Delta-Epsilon Proof of Continuity of a Function

Define $f\colon \mathbb{R} \times \mathbb{R}\to\mathbb{R}$ as $\dfrac{xy}{x^2 + y^2}$ for $(x, y) \neq (0, 0)$ and set $f(0, 0) = 0$. Determine whether $f$ is continuous. Please keep in mind that I'm ...
1
vote
3answers
53 views

Proof Using cartesian products

Suppose that $A$, $B$, and $C$ are sets. Prove that $(A\cap B)\times C =(A\times C)\cap(B\times C)$. Prove the statement both ways or use only if and only if statements.
3
votes
1answer
44 views

Induction proof for a summation: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$ [duplicate]

Prove by induction: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$. Hint: Use $k(k+1)^2 = 2(k+1)\sum i$. Basis: $n = 1$ $\sum_{i=1}^1 i^3 = \left[\sum_{i=1}^1 i\right]^2 \to 1^3 = 1^2 \to 1 = 1$. ...
1
vote
2answers
46 views

Proof by Induction that $16 \mid 5^n - 4n - 1$

Using induction, prove that $16\mid 5^n - 4n - 1$ for $n$ in $\mathbb{N}$ Here's what I have and what I'm stuck on: basis: $n = 1$, $5 - 4(1) - 1 = 0$ and $16\mid 0$. Hypothesis: Assume true for ...
2
votes
1answer
36 views

Help formalizing this proof about a continuous, one-one function.

I'm having a bit of trouble getting the language on this proof right, though I think I have the idea correct. I have the function $f\colon D \rightarrow {\bf R}$ where $D = [a,b]$. The function is ...
0
votes
1answer
51 views

Prove that there is a 1-1 correspondence between the set of subgroups of $\mathbb{Z}/N \mathbb{Z}$ and the set of the positive divisors of $N$

Im interested in the above Proof, is because I have the intiuition that it is not true at all, because for example, all the primes have exactly 2 positive divisors 1 an themselves, How Can I prove or ...
0
votes
4answers
40 views

Proof without using induction that a number is divisible by 6

Prove without using induction that all numbers of the form $6|8^n - 2^n$. I need a brush up on subtracting numbers with the same base but different exponent. So far I have $8^n - 2^n = 2^{3n} - ...
1
vote
3answers
78 views

If $f$ is continuous, $f(1) >1$ and $f(x+y)=f(x)f(y)$, then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
1
vote
1answer
79 views

Well defined Functions on Congruence classes

Could someone please confirm my logic or point me in the right direction? Thank you. 1) Is the function $f : [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = [n^2]_p$ well defined? 2) Is the ...