For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

learn more… | top users | synonyms

3
votes
2answers
95 views

Need to prove that “If $x+y \ge 1$ then $x \ge \frac 12$ or $y \ge \frac 12$”

So I have this one homework assignment where I have to prove the following clause "If $x+y \ge 1$ then $x \ge \frac 12$ or $y \ge \frac 12$". I thought that if I assign $x=y$ and put it like "$2x \ge ...
0
votes
1answer
47 views

Proof Involving Simple Graph

I have the task of proving that if there is a simple graph with 6 vertices and 13 edges, there is at least one vertex of degree greater than or equal to five. Given that, $2m = \sum_{v\epsilon V} ...
2
votes
3answers
95 views

Proving that a sequence is between certain values at certain n

I'm given that $a_1=1$, and for every $n \gt1, a_{n+1} = a_n + \frac{1}{a_{n}}$. I need to prove that $20 < a_{200} < 24$. I tried finding a limit at infinity setting both limits to $L$ ( for ...
2
votes
2answers
1k views

Accumulation Points & Convergence: A Sequence Existence Proof

Request: Prove that $x$ is an accumulation point of a set $S$ iff there exists a sequence $(s_n)$ of points in $S\setminus \{x\}$ such that $(s_n)$ converges to $x$. Attempt: Since this is a ...
3
votes
1answer
90 views

If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$.

Question: If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$. Prove or give a counterexample. Attempt: The statement is true because if $(\lvert ...
3
votes
3answers
186 views

Proving Limit False

I'm trying to prove that the limit of sin x as x->infinity is not equal to 1/2. I know that this is true, but I can't seen to figure out how to prove it using the precise definition of a limit. What ...
1
vote
4answers
134 views

Math inequality proof [duplicate]

If $a, b$ are positive real numbers and $a + b = 1$, prove that $$ \left(a +\frac{1}{a}\right)^2 + \left(b +\frac{1}{b}\right)^2 \geq \frac{25}{2} $$ Thank you.
1
vote
3answers
829 views

Limit of an $n$-th Root Proof

Prove that $\lim_{n \to \infty} a^{\frac{1}{n}} = 1$ if $a >0$. In my textbook, we are given a suggestion to let $a^{\frac{1}{n}} = (1+h_n)$ and then show that the $h_n$ term goes to zero using a ...
3
votes
1answer
149 views

Intro to proof in real analysis 1

This is what I have to prove: For elements $x, y$ in an ordered field, if $0 < x < y$ then $y^{-1} < x^{-1}.$ My proof: $0 < x < y$ multiply $x^{-1}$ on the left of both sides ...
0
votes
1answer
52 views

Prove that any completely regular semigroup $S$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$

Consider any completely regular semigroup $S$. I would like to prove, that any $a,b\in{S}$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$. So far, I was able to prove only the first ...
1
vote
0answers
51 views

Quantitative Economics: Continuity

How do I prove that $f(x)=e^x$ is a continuous function at the point $x=0$? I understand that anything raised to the $0$ power equals $1$, therefore it is continuous. But I don't know how to write a ...
0
votes
3answers
101 views

Math real numbers analysis.

Given a real number $x$, and a natural number $N\gt 1$. Consider the numbers : $0, x−\lfloor x\rfloor,2x−\lfloor2x\rfloor,\ldots,Nx−\lfloor Nx\rfloor.$ Show that some pair of these numbers differs ...
4
votes
0answers
125 views

Cauchy induction: are there examples of cases where choosing an integer other than $2$ is a better strategy?

Cauchy induction, sometimes called backwards induction, works as follows: show that $p(1)$ is true show that $p(n)$ implies $p(2n)$ (which inductively implies $p(2^n)$ is true) show that $p(n)$ ...
1
vote
1answer
643 views

Euler's Formula for Primes

Is there any way to prove that the Euler's Formula for Primes $n^2+n+41=41^2$ is valid? How would you even start to prove that a number is prime? If you could prove that a certain number is prime, it ...
6
votes
3answers
475 views

Proving statements by its contrapositive

Prove the following statement by proving its contrapositive: “If $n^3 + 2n + 1$ is odd then n is even” Therefore: $\lnot q \rightarrow \lnot p =$ "if $n^3 + 2n + 1$ is even then $n$ is odd. So ...
0
votes
1answer
1k views

The cardinality of the power set with $N$ elements is equal to $2^N$ [duplicate]

Let $\mathcal{P}(X_N)$ be the power set of a set $X$ with $N$ elements. I am trying to prove by induction that its cardinality $\mid \mathcal{P}(X_N) \mid = 2^N$. Firstly, I think it helps to ...
5
votes
1answer
68 views

A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
4
votes
1answer
83 views

The 2 Ways to Prove Uniqueness - Interchangeable or Nonidentical?

An element belonging to some prescribed set $A$ and possessing a certain property $P$ is unique if it is the only element of $A$ having property $P$. Typically, to prove that only one element of ...
2
votes
0answers
227 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
0
votes
1answer
250 views

Prove the set $S$ satisfies lub property iff it satisfies glb property

I've been stuck on this and although I have gone to professors for advice, I can't grasp it. Here I tried re-doing my proof and I can bet it is certainly wrong, but I am burnt out. I would like to ...
13
votes
4answers
702 views

Does proof by contradiction assume that math is consistent?

The standard proof by contradiction goes like It is known that $P$ is true. Assume that $Q$ is true. Using the laws of logic, deduce that $P$ is false. Rejecting this contradiction, we are forced to ...
5
votes
2answers
364 views

Ordering the field of real rational functions

Perusing the Wikipedia article on ordered fields, I encountered an interesting statement, a variation of which I am now trying to prove. Basically, I am trying to show that the set of real rational ...
2
votes
2answers
397 views

On the Divergence of $s_n=\cos{\frac{\pi}{3}n}$: A Proof

Question: Show that $s_n=\cos{\frac{\pi}{3}n}$ is divergent. Attempt: Suppose that $\lim_{n\rightarrow \infty}(\cos{\frac{\pi}{3}n})=s$, then given an $\epsilon$, say $\epsilon=1$, we can ...
1
vote
3answers
63 views

Uncertain how to proceed with combinatorics proof

The problem is as follows: let $n_1, n_2,..., n_t$ be positive integers. Prove that if $n_1+n_2+...+n_t-t+1$ objects are placed into $t$ boxes, then for some $i, i=1, 2, ..., t$, the $i$th box ...
2
votes
0answers
64 views

Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
2
votes
0answers
35 views

Verifying an integral identity related to asymptotic homogenization of an elliptic partial differential equation

Background I'm reading Hornung (1997)'s Homogenization and porous media, pg 3: We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, ...
2
votes
1answer
442 views

Prove that every real number belongs to the interval of integers

Here is a problem: Proposition. Prove that for every real number $x$, there exists a unique integer $N$ such that $N \leq x < N+1$. $Proof$ (Existence only). Since $x := \lim_{n\rightarrow ...
0
votes
0answers
2k views

Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational.

Question: Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational. Is your proof direct, by contradiction, or by contrapositive? State the converse. Prove or ...
1
vote
4answers
174 views

Prove of subsets [duplicate]

Prove that the number of subsets with odd number of elements is equal to the number of subsets with even number of elements. I'm not sure how to approach this problem. Is this even true for me ...
0
votes
1answer
44 views

How to find a function that meets a given condition?

Suppose I want to find a function $f$ that meets the following condition: $\dfrac{\partial f}{\partial x_i} . \dfrac{x_i}{f} = c$ where $f$ is a function of $x_i$. What is a systematic way to ...
0
votes
2answers
182 views

$S$ is a compact subset of $\mathbb{R}$ and $T$ is a closed subset of $S$ $\implies$ $T$ compact [duplicate]

If $S$ is a compact subset of $\mathbb{R}$ and $T$ is a closed subset of $S$, then $T$ is compact. How can I show this using the definition of compactness, and separately showing this by the ...
1
vote
2answers
108 views

“Collection”: What does it mean?

I've seen a lot of question of same ilk as the request I'm about to pose, but what I'd like to know is what does "any collection" mean in the following request: Prove that the intersection of any ...
3
votes
1answer
34 views

A question about $\varepsilon$-$N$ proofs.

Suppose I have an expression of the form $$(|a_m - a_{n+1}| + |a_n| + |a_m|)$$ where $(a_n)$ is a sequence convergent to $0$ and $m\le n$. I want to show that my expression is less than $\varepsilon$ ...
-1
votes
2answers
466 views

Bounded Infinite Set: Infinitely Many Points [duplicate]

How is it that if $S\subseteq\mathbb{R}$ is a bounded infinite set, where $x=\sup(S)$, then every neighborhood of $x$ contains infinitely many points of $S$?
5
votes
2answers
134 views

REVISTED$^2$: Fraction Existence Proof

Question 1: I'm asked to prove that there exists an $n\in\mathbb{N}$ such that $$\frac{1}{n+1}\leq\frac{a}{b}<\frac{1}{n},$$ where $0<\frac{a}{b}<1$. Here $\frac{a}{b}$ is a fraction in ...
1
vote
2answers
99 views

Let $B \subseteq A \subseteq \mathbb{R}^n$. Show that $B$ is closed relative to $A$ iff $B = A \cap C$ for some set $C$ closed in $\mathbb{R}^n$.

Let $B \subseteq A \subseteq \mathbb{R}^n$. Show that $B$ is closed relative to $A$ iff $B = A \cap C$ for some set $C$ closed in $\mathbb{R}^n$. My solution: We know $B$ is closed relative to $A$ ...
2
votes
1answer
43 views

$A \subseteq \mathbb{R}^n$ is closed iff for any $\{x_i\}_{i=1}^{\infty}$ in $A$ with limit $x$, we have that $x \in A$

PROBLEM: $A \subseteq \mathbb{R}^n$ is closed iff for any $\{x_i\}_{i=1}^{\infty}$ in $A$ with limit $x$, we have that $x \in A$. MY SOLUTION: Say $A$ is closed, and pick a sequence ...
8
votes
4answers
480 views

Idea of the Proof : Existence of a & b so that (Any integer greater than 8) = 3a + 5b [duplicate]

Claim: Prove that for every integer $n \geq 8$, there exist nonnegative integers $a$ and $b$ such that $n = 3a + 5b.$ Proclaimed solution : Let $n ∈ \mathbb{Z}$ with $n ≥ 8.$ $\text{ Then } n ...
1
vote
2answers
119 views

$\partial A $ is a closed set

Attempt: Pick a point $x \notin \partial A$, and Take a nghbd $N$ of $x$. If $N \cap \partial A = \varnothing$, then we are done. Suppose $y \in N \cap \partial A$. So, $y \in \partial A$, therefore ...
2
votes
1answer
32 views

Is a direct product $\prod_{\alpha\in{}A}S_\alpha$ of semigroups $S_\alpha$ simple, if all semigroups $S_\alpha$ are simple?

I am currecntly trying to give an answer to the following problem. Consider a family of semigroups $(S_\alpha)_{\alpha\in{}A}$ and let every semigroup $S_\alpha$ be simple. Is it true or not, that ...
0
votes
3answers
170 views

Are my proofs of elaborate consistency?

I have to complete some proofs for homework and here are my attempts. I have reviewed my proofs many times such that there would be no erroneous symbols used, to my best efforts. For my first proof I ...
2
votes
6answers
215 views

is $\{(x,y) : x,y \in \mathbb{Z} \}$ a closed set?

I claim yes, and to show this, it will suffice to show that $\mathbb{R}^2 \setminus \mathbb{Z}^2$ is open. So that for every $x \in \mathbb{R}^2 \setminus \mathbb{Z}^2$, we must find a neighborhood ...
0
votes
1answer
60 views

proof, for a differentiable function from $\mathbb{R}^n$ to $\mathbb{R}$, $\int_{Cpq}{\nabla f\cdot\mathrm{d}\boldsymbol{r}} = f(q)-f(p)$

For a differentiable function: $f:\mathbb{R}^n\rightarrow\mathbb{R}$ prove that: $$\int_{C_{\boldsymbol{pq}}}{\nabla f}\cdot\mathrm{d}\boldsymbol{r}=f(\boldsymbol{q})-f(\boldsymbol{p})$$ where $C$ ...
0
votes
1answer
92 views

Recursive fibonacci algorithm correctnes? [proof by induction]

im studying for the computer science GRE, as an exercise i need to provide a recursive fibonacci algorithm and show its correctness by mathematical induction. here is my recursive version of ...
1
vote
1answer
431 views

Proving the transitivity of a relation

I want to prove that the relation $\sim$ on fractions given by $\frac{a}{b} \sim \frac{c}{d}$ if $ad = cb$, where $a, c \in \mathbb Z$ and $b, d \in \mathbb Z_{> 0}$, is transitive. (My last ...
1
vote
1answer
55 views

Prove that $a_n \times b_n \to 0$ for $n \to \infty$

I want to prove this example: If $a_n \to 0$ for $n \to \infty$ and $(b_n)_n$ is bounded. Prove that $a_n \times b_n \to 0$ for $n \to \infty$. My first guess is that I should use the definition ...
1
vote
1answer
120 views

The union of a collection of open sets is open

My attempt: Say $\{A_{\alpha}\}$ is a collection of open sets. If we can show that $\operatorname{Int}(\bigcup A_{\alpha}) = \bigcup A_{\alpha}$, then we are done. We know the interior of any set is ...
0
votes
5answers
120 views

$\mathbb{R}^2 - \{x\mbox{-axis}\}$ is open

problem: $\mathbb{R}^2 - \{x\mbox{-axis}\}$ is open My attempt: We want to show that there exists a neighborhood $D(x,r)$ around $x$ such that $D(x,r) \subseteq \mathbb{R}^2 - \{x\mbox{-axis}\}$. In ...
-2
votes
3answers
90 views

Prove that this inequality $5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$

Let $a,b,c>0$ and $a+b+c=1$ Prove that $$5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$$
3
votes
0answers
61 views

How does Hildebrands proof of the prime number theorem via large sieve work?

How does the sieve inequality (I may not know the most general form) lead to the distribution of primes? To me, these concepts do not seem to be related. Can their connection be described in a ...