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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3answers
55 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
110 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
18 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
62 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
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3answers
41 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
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0answers
11 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 ...
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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 $B=\{d>0: divisor of N\}$, ...
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0answers
13 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
38 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
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1answer
24 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
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1answer
32 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
44 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
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2answers
31 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.
2
votes
1answer
25 views

Induction proof for a summation

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
40 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
31 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
41 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
38 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
72 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 ...
0
votes
1answer
41 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 ...
1
vote
2answers
34 views

Show $f(rx) = [f(x)]^r$ where $r\in\mathbb{Q}$.

Consider the function $f$ with the following properties: $$(1) \lim_{x\rightarrow 0} f(x) =1,$$ $$(2) f(x+y)=f(x)f(y),$$ $$ -\infty<x,y<\infty.$$ Show that $f(rx)=[f(x)]^r$ where ...
1
vote
2answers
52 views

Proving directly that ($a+b)^3 \equiv a^3 + b^3 \mod 3$

Assuming a and b are integers, I must prove directly that: $$ (a + b)^3 \equiv (a^3 + b^3) \mod 3 $$ First, my peers and I made the mistake of assuming what we are trying to prove and thus failed. ...
0
votes
1answer
46 views

Limit proof for rational function $\frac{1}{x}$

A while ago I posted another one like this with a incorrect approach, please see this one! Is this an accurate proof for limits for the function $\frac{1}{x}$ $\displaystyle \lim_{x\to1} \frac{1}{x} ...
0
votes
1answer
46 views

Proof of Functions!

Question: Let $ f\colon Z \to Z $ and $ g\colon Z \to Z $ be two functions. Prove that the following are functions. a. $h(x)\colon Z \to Z $ defined as $h(x) = f(g(x))$ when $g(x)$ is an onto ...
0
votes
1answer
49 views

Equivalence classes of multiples of 3

I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition: for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3. 1) List four ...
3
votes
1answer
29 views

Eulerian circuit with no isolated vertex is connected

This is my first question (ever), and I am pretty new to math. So I ask for patience and understanding in advance. So this is the proof I came up with: Consider $G = (V,E). $ By definition of ...
1
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2answers
33 views

Well-defined functions on equivalence classes

Could someone please explain how to develop a proof and the reasoning behind the following: My professor said that to determine if a function is well-defined, we check to see if equivalent elements ...
3
votes
2answers
86 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
0
votes
1answer
30 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
0
votes
0answers
28 views

Show that both $*$ and $.$ operation are same.

Let $G$ be a topological group with identity $x_0$. Let $\pi_1(G,x_0)$ is a fundamental group with the usual $*$ operation. If we define $(f.g)(s)=f(s)g(s)$ $\forall s\in [0,1]$ $\forall f,g\in ...
1
vote
0answers
32 views

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \to \sqrt{x}$

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \rightarrow \sqrt{x}$ for $x > 0$. Let $\epsilon > 0$ be arbitrary, want to find $N \in \mathbb{N}$ such that $n \geq N \Rightarrow ...
0
votes
2answers
59 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
0
votes
1answer
29 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum ...
3
votes
1answer
52 views

Show that if A is diagonalizable, then sin^2(A) + cos^2(A) = I. Does this identity also hold for nondiagonalizable matrices?

Show that if A is diagonalizable, then $\sin^2(A)+\cos^2(A)=I$. Does this identity also hold for nondiagonalizable matrices? This is what I got so far: $$ e^{iA}= \cos A +i\sin A \\ \cos A= ...
1
vote
1answer
63 views

Is this an accurate limit proof for sine?

$\displaystyle \lim_{x\to 0} \frac x{1 + \sin^2(x)} = 0$ proof $$\left|\frac x{1 + \sin^2(x)}\right| < \epsilon$$ $$|x| < \delta$$ Let's require $|x| < 1$ so therefore, $$\sin^2(|x|) + 1 ...
0
votes
1answer
45 views

Limit proof for $1/x$ (as $x \to 1$)

Prove $\lim_{x\to 1} \frac{1}{x} = 1$ Using $\epsilon-\delta$ $|\frac{1}{x} - 1| < \epsilon$ for some $|x - 1| < \delta$ $|\frac{1}{x} - 1| = \frac{|1-x|}{|x|}$ Lets require $|x - 1| < ...
0
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1answer
17 views

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$ I want to do the $n - 1 \rightarrow n$ induction step. But I'm confused as to what my base case is. Usually if I want ...
1
vote
3answers
28 views

Prove that a sequence has a limit

Sequence $a_{n}$ satisfies $|a_{n}| \leq n$ for all $n \in \mathbb{N}$. Let sequence $b_{n} = \frac{a_{n} + 5}{n^{2} + a_{n}}$, prove that $b_{n}$ has a limit, and find it. I know that $b_{n}$ has a ...
1
vote
1answer
53 views

How to interpret the logic of an “or” in a matrix proof.

I am trying to learn to better interpret the meaning of equations and that is the purpose of this question, not just to find the proof, but to find the logical flow of the proof and understand it. I ...
1
vote
2answers
27 views

Next step in proof of sets

Proposition to prove : (A-B)∩(B-A) = 0 So, I understand why this is 0, I'm just not sure what propositions should be used in proving so. I have this so far 1)(A-B)∩(B-A) :Premise ...
3
votes
1answer
26 views

If $H,K$ are subgroups of $G$, and $G$ is finite, prove that $[K\colon (H\cap K)]\leq [G\colon H]$

Let $H,K$ be subgroups of a finite group $G$. Prove that $[K\colon (H\cap K)]\leq [G\colon H]$. This is what I have: $[K\colon (H\cap K)] = |\left\{ a(H\cap K) \mid a\in K\right\}|$ $[G\colon H] = ...
0
votes
1answer
61 views

If a sequence ${a_n}$ is monotonically increasing. then $\lim_{n \to \infty} a_n = \sup{(a_n)}$

Can you please tell me if my proof is correct: If a sequence ${a_n}$ is monotonically increasing. Then $$\lim_{n \to \infty} a_n = \sup{(a_n)}$$ Proof: $$a_n\leq a_{n+1}\leq \sup(a_n)$$ Assume ...
0
votes
2answers
25 views

Proof by induction with variable other than $n$

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
2
votes
1answer
54 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
2
votes
2answers
44 views

Show that $\int_{x=a}^{x=b} f'(x) g(x) dx=f(b)g(b)-f(a)g(a)-\int_{x=a}^{x=b} g'(x)f(x)\, dx$

I have to prove the following: Suppose $f$ and $g$ are differentiable on $[a,b]$ and $f'$ and $g'$ are integrable on $[a,b]$. Prove that $f'g$ and $g'f$ are integrable on $[a,b]$ and that of: $$ ...
1
vote
4answers
64 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
6
votes
2answers
46 views

Proving a complete and totally bounded metric space is compact.

I'm having trouble writing down the details of this proof formally. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, ...
1
vote
0answers
31 views

How to prove unicity in a disjunction of $n$ propositions

Let's suppose I have the propositions $\varphi_1, \varphi_2,...,\varphi_n$ and I want to prove that there happens exactly one of them. How do you do it? To do it the hard way I guess we first need to ...
1
vote
6answers
65 views

Prove that $gcd(a, b) = gcd(b, a-b)$

I can understand the concept that $\gcd(a, b) = \gcd(b, r)$, where $a = bq + r$, which is grounded from the fact that $\gcd(a, b) = \gcd(b, a-b)$, but I have no intuition for the latter.
0
votes
1answer
38 views

There exists a positive real number $u$ such that $u^3 = 3$

Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$. Show that there exists a positive real number $u$ such that $u^3 = 3$. So far, I have come up ...