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

0
votes
1answer
51 views

Intro to proofs: continuity of f

Prove that every continuous function f can be written f = g−h, where g and h are nonnegative and continuous. If I define g=max{f,0} and h=max{0,-f} Do I need to prove that g and h are still ...
3
votes
9answers
71 views

Prove that $a+1 < a^2$ for all integers $a > 1$

I know it is true, but how could I prove it? $$a^2-(a+1)>0$$ $$a^2 - a -1 >0$$ via a graphical solution $a^2-1-1>0$ when $a>$ approx $1.68$...thus given $a$ is an integer $>1$ the ...
3
votes
1answer
53 views

Find supremum and infimum of $A=\left\lbrace\frac{2013}{1+\epsilon+\epsilon^{-1}}:\epsilon\in(0,1)\right\rbrace$

Find $\sup{A},\inf{A},\max{A},\min{A}$ where: $$A=\left\lbrace\frac{2013}{1+\epsilon+\epsilon^{-1}}:\epsilon\in(0,1)\right\rbrace$$ I suspect that $\sup{A}=\frac{2013}{3}, \inf{A}=0$ and max and min ...
2
votes
5answers
121 views

Prove $ \frac {x_1 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$

I am studying computer science in the first term. I have to proof the following inequality: $$ \frac {x_1 + \cdots+ x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$$ $x$ can be any positive real ...
5
votes
1answer
83 views

How to prove that this function will converge to 1?

I have an assignment for tomorrow that ask me to prove that the sequence/function f(x) = x/2 if x is even or 3x+1 if x is odd (where x is a natural number) will converge to 1. I have tried by hand ...
-1
votes
2answers
57 views

Prove that $\lim_{x\to a}e^x=e^a$

Prove by epsilon-delta limit definition that: $\lim_{x\to a}e^x=e^a$ My definition of exponential function is $e^x=\lim_{n\to \infty}(1+x/n)^n$. My teacher said that we need to use it but when I ...
0
votes
0answers
28 views

If $f,g$ are integrable on $[a,b]$, then $h(x)=max{f(x),g(x)} $ is Riemann integrable.

I have to prove the following: Suppose $f,g$ are Riemann integrable on $[a,b]$. Define $h(x)=max\{f(x),g(x)\}$. Prove that $h$ is Riemann integrable on $[a,b]$. My attempt: If the case is that ...
0
votes
2answers
28 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 ...
2
votes
1answer
57 views

Proving $M$ is maximal if the quotient ring $R/M$ is a field.

Let $R$ be a ring with unit element and ideal, $M$, such that $R/M$ is a field. Prove $M$ is maximal ideal. I know that because $R/M$ is a field, its only ideals are $(0)$ and itself. Also, I ...
1
vote
1answer
12 views

Maximizing with two arguments with iteration

I would like to show that: $$ \displaystyle\max_{(x,y) \in X\times Y} f(x,y) = \displaystyle\max_{x} \displaystyle\max_{y} f(x,y) $$ which I obviously believe to be true, although I may be wrong. ...
0
votes
4answers
106 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 ...
2
votes
2answers
48 views

Prove that if $p$ is a prime and $p|k^n$, then $p^n|k^n$

I want to prove that if $p$ is a prime and $p|k^n$, then $p^n|k^n$ but I have no idea where to start.
0
votes
2answers
22 views

True of False: If $f(x)\to 0$ as $x\to a+$ and $g(x)\ge 1$ for all $x\in \mathbb R$, then $g(x)/f(x)\to \infty$ as $x\to a+$

True of False: If $f(x)\to 0$ as $x\to a+$ and $g(x)\ge 1$ for all $x\in \mathbb R$, then $g(x)/f(x)\to \infty$ as $x\to a+$. I am having trouble trying to prove how this could be true (as I can't ...
1
vote
2answers
45 views

How to prove by definition (epsilon-delta) this limit of 2 variables?

I have this limit $$\lim_{(x,y)\rightarrow(1,0)}x^2\cos(y)=1$$ How to prove by definition: $$\lim_{(x,y)\rightarrow(1,0)}x^2\cos(y)=L\Longleftrightarrow\forall\varepsilon>0,\exists\delta>0\ ...
2
votes
2answers
21 views

if $Re(z\overline w)=|z||w|$ then $w=tz$ where $t\in \mathbb R$

I was given the following exercise in complex numbers: Assume that for some $z,w \in \mathbb C$: $Re(z\overline w)=|z||w|$ Show that this implies $w=tz$ where $t$ is some positive real. What I ...
0
votes
0answers
19 views

Fermat's little Theorem with a smallest postitive integer [duplicate]

Let $p$ be a prime and $a$ be an integer such that $p\not| a$. Suppose $d$ is the smallest positive integer such that $a^d\equiv1$ (mod $p$). Prove that $d|(p-1)$. I already know that the first ...
6
votes
1answer
27 views

Explaining the purpose of the remaining part of this proof .

I'm given that:= $E\subseteq \mathbb R^n$ be open and $f:E\to \mathbb R^n$ be a $C^1$ map . Suppose that for some $a\in E$ , the linear map $f'(a)$ is invertible ,and $b=f(a)$ .Then := I've to show ...
1
vote
1answer
62 views

Proof: How many edges need be removed from this graph to produce the spanning tree?

Assume the graph,$G$ has the degree sequence $6,4,4,3,3,3,3,2,2$. How many edges must be removed from $G$ to produce the spanning tree $T$? We can construct this graph using Havel-Hakimi's ...
1
vote
2answers
47 views

Intro to Proofs: Continuity

Let $c > 0$ and $f : \mathbb{R} → \mathbb{R}$ satisfy $$|f(x) − f(y)| ≤ c|x − y|$$ for all $x, y ∈ \mathbb{R}$.Show that $f$ is continuous. Does showing that $|x − y|≤ \frac{\delta}{c}$ and then ...
0
votes
1answer
26 views

$T(n)=T(n-1)+O(\log n)$ is $T(n)=O(n^2)$ or $T(n)=O(n \log n)$

I have this Recurrence relation: $T(n)=T(n-1)+O(\log n)$ What is the solution? $T(n)=O(n^2)$ or $T(n)=O(n \log n)$ What I did is: I assume that $T(n)\le O(n^2)$ And that's bring me to $O(n^2)$, ...
1
vote
1answer
48 views

Application of Reflection Principle for Holomorphic functions

Let $f$ be holomorphic on $D'(0,1)=\{0<|z|<1\}$ and $f$ is continuous and real valued on $\{|z|=1\}.$ Show $f$ can be extended to $\mathbb{C}-\{0\}$ such that $f(z)= ...
0
votes
1answer
18 views

Continuity of single variable functions defined in terms of definite integral

Let $f$ be a continuous function defined on $[a,b] \times [c,d]$ and $g(x)=f(x,y_o)$ is a continuous function defined on $[a,b],$ for each $y_o \in [c,d]. $ Could anyone advise me how to ...
2
votes
2answers
44 views

Prove that g(y)>0 for all y in the real numbers

Let g:$\mathbb{R}\to \mathbb{R}$ such that ($i$) for all $y_{1},y_{2} \in \mathbb{R}$, $g(y_{1}+y_{2}$)=$g(y_{1})g(y_{2})$ suppose in addition that ($ii$) there exists $y \in \mathbb{R}$ such ...
1
vote
1answer
51 views

Introduction to Proofs: Supremum and Infimum

(b) Find the supremum and infimum of the set $S_a = \{a/(a−x) \mid x \in (0, a)\}$ for some $a>0$ and prove that you have found them. For the supremum, I said that it did not exist, because the ...
1
vote
0answers
6 views

Proof of variance

I came across a question in a book titled "An introduction to Generalized Linear Models" which is given below: I need the solution to question (b). So, I tried adding −μ and + μ, but could not find ...
0
votes
1answer
26 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 ...
2
votes
3answers
48 views

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

Prove: Use the triangle inequality to prove that for all $x, y, z, |x-z|≤|x−y|+|y−z|$ Is my proof correct? Proof: Let $a = x-y$, and $b=y-z$. We can say that $|a+b| = |(x-y) + (y-z)| ...
1
vote
2answers
74 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 $x ≤ 0$ ...
3
votes
2answers
76 views

Intro to Proofs: Proving Distance

Suppose that $f$ is continuous on $[a, b$], and let $α \in \mathbb{R}.$ Prove that there is a point on the graph of f which is closest to $(α, 0)$, i.e. $\exists y ∈ [a, b]$ such that the distance ...
0
votes
1answer
31 views

Proving a complicated statement about supremum of a bounded set - how to proceed?

I'm trying to solve the following problem: Say whether this statement is true for every $A\subset\Bbb{R}$ bounded from above ($A\not=\varnothing$) and for every $d\in\Bbb{R}$: $$[(\sup{A} = d\notin ...
0
votes
2answers
23 views

Show that $f(b)=\sum_{r=0}^n \frac{f^{(r)}(a)}{r!}(b-a)^r + \frac1{n!} \int_a^bf^{(n+1)}(t)(b-t)^ndt$

Let $f^{(n+1)}$ be integrable on [a,b]. Show that $$f(b)=\sum_{r=0}^n \frac{f^{(r)}(a)}{r!}(b-a)^r + \frac1{n!} \int_a^bf^{(n+1)}(t)(b-t)^n \ dt$$ From the form it looks like I'll need to use ...
3
votes
1answer
37 views

Symmetric groups and matrices

I am currently working through this question. I have completed part (a) and (c), however I am unable to make any progress with (b). I know $S_n$ is the symmetric group on n symbols, and that it has ...
1
vote
1answer
61 views

Application of Cauchy Integral

Let $f$ be holomorphic in $\{|z|<R\},$ where $R>1.$ Show: $\begin{align}f(z)= i\text{Im}f(0) +\dfrac{1}{2\pi} \int^{2\pi}_{0} \dfrac{e^{it}+z}{e^{it}-z} \text{Re}f(e^{it})dt, \ \forall |z| ...
-1
votes
1answer
30 views

Proving a certain set is inductive?

Let $m$ be a natural number in a field $F$ and let $$ S_m= \{k:k\in N ~~~and~~~ k\leq m \}\cup\{x:x\in F, m<x\} $$ Show that the set $S_m$ is inductive. Thanks in advance!
1
vote
1answer
30 views

Prove that g(0)=1 based on two conditions

Let g:$\mathbb{R}\to \mathbb{R}$ such that (i)for all $y_{1},y_{2} \in (0, \infty)$, $g(y_{1}+y_{2}$)=$g(y_{1})g(y_{2})$ suppose in addition that (ii) there exists $y \in \mathbb{R}$ such that ...
1
vote
1answer
187 views

Number of roots of a polynomial over a finite field

For any $g$ in $\mathbb{Z}/p\mathbb{Z}[x]$ prove that the degree of $f = \gcd(x^p - x, g(x))$ is exactly the number of distinct roots of $g$ in $\mathbb{Z}/p\mathbb{Z}$. My main problem is that I ...
2
votes
0answers
31 views

Accumulation Point Proof Topology Help

Show that the following definitions are equivalent (both ways). Def 1: $x$ is an accumulation point of $S$ if for all $\delta > 0$, $[(x - \delta , x) (x , x + \delta)] \cap S \not = \emptyset$ ...
4
votes
4answers
76 views

How to prove $n!\leq(\frac{n+1}{2})^n$ [duplicate]

Prove that for $n\in\mathbb{N}$ $$n!\leq(\frac{n+1}{2})^n$$. I'v solved base case for $n=1$ $$1\leq(\frac{1+1}{2})^1=1$$ The second step I've mada was that I assumed that $n!\leq(\frac{n+1}{2})^n$ And ...
0
votes
0answers
7 views

Need help understanding explanation about conjugations

So my problem asks me to find all subgroups of S3 that are conjugate to {$\rho_{0}, \mu_{2}$}. The answer key that I am looking at says: "To find all subgroups conjugate to {$\rho_{0}, \mu_{2}$}, we ...
0
votes
1answer
23 views

Properties of Harmonic functions and Log

Could anyone advise me on how to prove: If $g$ and $\text{log}|g|$ are harmonic in a simply connected domain $\Omega$, then $g \equiv$ constant on $\Omega.$ Hints will suffice, thank you very much.
0
votes
2answers
84 views

Prove $-(-a) = a$

Let $F$ be a field and $a \in F$. Prove $-(-a) = a$. So we want to show that $(-a) + (-(-a)) = 0$, since inverses are unique (I successfully proved that inverses are unique in an earlier problem ...
0
votes
1answer
25 views

Pre-images of functions, subsets

Let f:X→Y be a function.Consider sets A⊆Y and B⊆Y.Show that if A⊆B,then f^(-1)(A)⊆f^(-1) (B) I have assumed A⊆B and let x be arbitrary in f^(-1)(A), and know that the statement is true but it seems ...
0
votes
0answers
17 views

Prove that a certain (involving integrability) sequence converge to $\int_{0}^{1} f(x) dx$

I have to prove the following result: Suppose $f$ is Riemann integrable on $[0,1]$.Define: $$a_{n}= \frac{1}{n} \sum f (\frac{k}{n})$$ For all $n$. Prove that ${a_{n}}$ converges to $\int_{0}^{1} ...
2
votes
3answers
60 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 ...
2
votes
1answer
41 views

Group Theory: Isomorphisms

Let $H$ be a subgroup of the group $G$. Claim: $S=\{aHa^{-1} : a \in G\} \cong H$. My sketch: Define $\phi : S \to H$ by $aha^{-1} \mapsto h$. For injectivity, suppose $\phi(x) = \phi(y)$. By ...
1
vote
2answers
76 views

Conditional probability problem about dependent events?

Sorry about the not-so-descriptive title. Here's the problem: "There are $n$ coins arranged on a table, and $m$ of them are heads. You randomly select $p$ coins to remove one by one, but you forget ...
3
votes
1answer
204 views

Suppose that every vertex of $G$ has degree at least 3. Prove that $G$ has a cycle of even length.

I've been working through some graph theory problems and recently encountered one which had me stumped. Fortunately, a solution was provided by my resource. Unfortunately, the solution does not seem ...
-1
votes
2answers
47 views

F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?
1
vote
2answers
51 views

Proof if $x>0$, $\,x^{1/n}$ tends to $1$

I am looking for a proof possibly using the sandwich theorem and/or Bernoulli's inequality for proving the following statement: If $x>0$ then $x^{1/n}$ tends to 1. Thanks in advance.
0
votes
1answer
19 views

How to prove that at Complete Binary Tree (CBT) at height $h$ we have $2^h$ leaves

I try to prove it by induction, please tell me if I'm right... The induction assumption - For every CBT at height $h$ there is $2^h$ leaves. The base of the induction is right (I'm writing this proof ...