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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0
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1answer
89 views

How to prove a harmonic function is a polynomial?

Given $u(x)$ be entire harmonic function on $\mathbb R^n$ , and satisfy $u(x)\geq-C \left(1+|x|\right)^m$, where $C$ and $m$ are constants. ($m$ is an integer) Prove $u(x)$ is a polynomial of degree ...
0
votes
2answers
57 views

Why does this limit work?

Let $h(x)= (1+1/x)^x$ and $g(x)$ be another function. Now suppose $\lim\limits_{x \to \infty} g(x)= \infty$. Then $\lim\limits_{x \to \infty} h(g(x))$ =$\lim\limits_{x \to \infty} h(x)=e$. I would ...
4
votes
1answer
148 views

For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?

If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$? Please help me to clarify the above. Thanks in advance.
4
votes
2answers
108 views

If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
0
votes
1answer
118 views

Almost surely equal random variables and expectation

Let $X, Y$ be random variables defined on the same probability space $(\Omega, \mathcal{F}, P)$. I'm interested in seeing a proof for the following results: a) If $P(X = Y) = 1$, then $E(X) = E(Y)$. ...
2
votes
2answers
79 views

Proving $\lfloor 2x \rfloor = \lfloor x \rfloor + \lfloor x+0.5\rfloor$

I can intuitively see that this is true, but I'm having a very hard time proving it. I'm actually not even quite sure where to begin. I tried using the inequalities that define the floor function, and ...
1
vote
2answers
636 views

Check if this proof about real numbers with an irrational product is correct.

Can anyone confirm if my proof is correct, please? Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” Proof:- Assume that both ...
5
votes
2answers
185 views

Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
1
vote
1answer
368 views

Subsequence of $\sin n$

Find one convergent subsequence of $a_n= \sin n$ and prove it. I know that for all $n_1<n_2<n_3<\dots$, then $a_{n_1},a_{n_2},...$ is the sub-sequence of $a_n$. But I don't know how to ...
1
vote
3answers
712 views

How to write this in mathematical notation?

I have the following claim: “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” I'm supposed to write this in mathematical notation. It's ...
1
vote
2answers
404 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
0
votes
1answer
58 views

Operations between Subsequences' proof

Let $A$ and $B$ be non-empty bounded subset of $R$ a) If $C={x+y:x∈A,y∈B}$, prove that $C$ is bounded above and that $\sup ⁡C=\sup⁡ A+\sup ⁡B$. b) If$ D={x-y:x∈A,y∈B}$, prove that $D$ is ...
0
votes
1answer
51 views

Consequence of completeness axiom proof

Prove that , for a set $A⊂R$, $s=sup⁡A$ if and only if i) $a≤s$ for all $a∈A$. ii) For any $ε>0$, there exists $a∈A$ such that $a>s-ε$ I can easily prove that if $s=sup⁡A$ then $a≤s$ for all ...
0
votes
1answer
76 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
0
votes
2answers
64 views

Proof that a function is surjective to R

I'm having difficulties proving that the function $$\frac{\sin(\frac1x)}{x^2}$$ is surjective to $\mathbb R$. on the interval $(0,10]$. I tried to use the intermediate theorem, but that of ...
3
votes
1answer
46 views

Proving that a $p$-group operating on a finite set of order not divisible by $p$ has a fixed point.

Let $G$ be a finite group of order $p^e$ for some prime $p$. Let $S$ be a set of size not divisible by $p$. I know that $|G| = |Stab(s)|\cdot|O_s|$ = (stabilizer of $s$)(orbit of $s$) So if there ...
1
vote
3answers
245 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
0
votes
2answers
75 views

Monotone subsequence proof

prove that " Prove that every sequence a_n has a monotone sub-sequence." I tried to prove this by a proof of contradiction, so I assume there exist a sequence that doesn't have monotone sub-sequence, ...
4
votes
2answers
190 views

limit superior and limit inferior proof

$$\limsup \left(\frac 1{a_n} \right)=\frac 1{\liminf(a_n )} $$ I know this is true base on the definition of $\limsup$ and $\liminf$, but I don't know how to prove it formally.
1
vote
3answers
108 views

lemma of density of real number proof

If $a$ and $b$ are two distinct real numbers and $α$ is an irrational number, prove that there exists a rational number $r$ such that $rα$ lies between $a$ and $b$. I know that by the density of real ...
1
vote
1answer
66 views

Supremum of a set proof.

Let $a$ be a real number and let $S={x\in\Bbb Q:x<a}$. Prove that $a=\sup⁡(S)$. This obvious that $S$ is bounded above by $a$, and I know that by the completeness axiom, the least upper bound of ...
5
votes
1answer
49 views

Is this a valid proof of the contrapositive?

The question is the following: if $a$ and $b$ are distinct group elements, then either $a^2 \neq b^2$ or $a^3 \neq b^3$. I find this difficult to prove directly, so I formulated the contrapositive to ...
0
votes
1answer
567 views

Lim Sup and Lim Inf

This is a question with a few components to it. In each one, I give my attempt at a solution. Thank you in advance for any suggestions/answers/advice on how to solve this problem. For bounded ...
0
votes
0answers
71 views

In this insertion sort algorithm for example, how would I prove the algorithm's time complexity is O(n^2)?

Take the following insertion sort algorithm: I know it's O(n^2) fairly easy by examining it. But as far as proving it's O(n^2), how would I go about doing that? I could add up all the operations, ...
2
votes
2answers
79 views

Absolute Convergence

Prove that if $\vert \frac{a_{n+1}}{a_n} \vert \leq \vert \frac{b_{n+1}}{b_n} \vert $ for $ n >>1$, and $\sum b_n$ is absolutely convergent, then $\sum a_n$ is absolutely convergent. My ...
1
vote
2answers
29 views

Help with an induction proof

Here's the thing I'm trying to prove $2^k - (2^{k-1} + 2^{k-2} + ... + 2^2 + 2^1) = 2$ It's obviously easy for the k = 1 case, but I'm stuck on the k + 1 case.
0
votes
1answer
108 views

How do I prove an algorithm has $n^3$ time complexity?

Take the CYK algorithm outlined here: How to prove CYK algorithm has $O(n^3)$ running time In the top answer, how did that person go from the three summations to $t=(n^3−n)/6$ ? What's the method ...
4
votes
4answers
563 views

Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$ [duplicate]

Prove that $$\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}.$$ This problem is in the chapter about discrete random variables, but I have no idea what to go about substituting. I can't ...
0
votes
2answers
894 views

Prove that $n+1 \choose k$ = $n \choose k$ + $n \choose k-1$

As the title says. Prove that $n+1 \choose k$ = $n \choose k$ + $n \choose k-1$ It looks to me like induction but since there are two variables, I'm not really sure how to even set up a base case. ...
0
votes
1answer
48 views

Is simple straight-edge and compass construction a substantial proof?

I'm working on a problem that asks to prove that a point $D$ is outside of a $\triangle ABC$, on the circle through the triangle, given that sides $AB$ and $AC$ are not congruent, and that $D$ is the ...
5
votes
5answers
219 views

Show that $\gcd(a,b)=\operatorname{lcm}(a,b)$ if and only if $a=b$.

I know how to prove $a=b$ only if $\gcd(a,b)=\operatorname{lcm}(a,b)$, but I don't know how to prove the "if part". Can anyone help me?
1
vote
1answer
36 views

can we interpret a sentence involving more than one 'iff' in the following way?

If there are a sequence of 'iff' in a sentence, can we make a conclusion from the sentence by dropping all the 'iff' that lie after the first 'iff' and drop all the statements between the first ...
3
votes
1answer
38 views

Proofs as games?

A long time ago (but I can't remember when), I was introduced to the (pedagogical) concept of writing a proof as giving a winning strategy for a game. Basically, given a statement $\forall x\exists y ...
0
votes
3answers
85 views

$g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.

I'm trying to write a proof for this, but I don't know how to begin. Any help appreciated. Suppose $g$ is a function such that $g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that ...
0
votes
2answers
55 views

Linearly independent matrix proof

Suppose that $S=\{u_1,u_2,…,u_n\}$ is a set of vector from $\mathbb{R}^m$. Show that $S$ is linearly independent if and only if the set $S'=\left\{u_1,\ \sum_{i=1}^2 u_i,\ \sum_{i=1}^3 u_i,\ \ldots,\ ...
5
votes
1answer
114 views

showing $a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$ is not Cauchy

My gut telling me that the following sequence is not Cauchy, but I don't know how to show that. $$a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$$
0
votes
2answers
50 views

Statement of a lemma

Suppose $\alpha$ and $\beta$ are sets. Suppose that the formula $\alpha\supseteq\beta$ is almost obvious. Now which of the following (true) alternatives make a statement of a lemma (used by one ...
0
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2answers
58 views

how to prove this: $f(A)=B$

I am given two sets: $A$ and $B$ and a function $f: A \rightarrow B$. I am asked to show and prove whether $f(A)=B$ is true or false. I am stuck not knowing how to do this. How can I do this?
3
votes
1answer
58 views

Rephrase the proof “For all odd n, there exists a Group G …” [duplicate]

I am trying to construct a proof and would like to know if I have started it correctly. The proof is as follows. "Prove that for every odd integer n, there is a group with exactly n elements of order ...
4
votes
3answers
132 views

Increasing and bounded sequence proof

Prove that the sequence $a_n= 1+ \frac 12+ \frac 13+\cdots+ \frac 1n-\ln(⁡n)$ is increasing and bounded above. Conclude that it’s convergent. This what I got so far Proof: Part 1: Proving $a_n$ is ...
0
votes
2answers
361 views

Cauchy sequence proof

Let $0<r<1,M>0,$ and suppose that $a_n$ is a sequence such that , for all $n∈N, |a_{n+1}-a_n |≤Mr^n$. Prove that $a_n$ is Cauchy. I know that I need to show for all $ε>0$, there exists an ...
4
votes
2answers
338 views

Smart Pen For Math Writing

Next month I will begin to learn in the university, and I am not sure if to buy smart pen such as Livescribe or Logitech IO 1/2 to write math with.(handwriting is not an option) The problem is that I ...
1
vote
1answer
106 views

How is it proved that S is dense in $\mathbb R$

This question is from here. The question is: But, I can't understand how is it proved that S is dense in $\mathbb R$. I also can't understand how "dense in $\mathbb R$" and "dense in $[0,1]$ ...
1
vote
1answer
76 views

Proof Checking and input: Generators of $\mathbb{Z}_{pq}$

I'm self-studying abstract algebra (slowly but surely), and I have a question about my answer to the following prompt: Problem statement: Show that there are $(q-1)(p-1)$ generators of the group ...
0
votes
1answer
32 views

Proof of Complexity

If I want to prove that $f=O(g)$ for $f(x)=x^{1/2}$ and $g(x)=x^{2/3}$, is it sufficient to say that $\lim_{x \to \infty}f(x)/g(x)=0$? I'm not sure if this is a convincing enough argument or more ...
2
votes
5answers
320 views

Prove $e^{\ln{x}} = x$

Is it possible to prove $e^{\ln{x}} = x$ for a student or can you only say that exponentiation is defined to be the inverse of natural logarithm and stop there?
2
votes
1answer
281 views

Proof involving angle bisector in an arbitrary triangle

In the above figure, AD is a bisector angle A (angle BAC). How do I prove in a triangle ABC of any dimensions that, $AB > BD$ $AC > CD$ Is it also possible to prove that, $AB > AD$ ...
0
votes
1answer
55 views

$S$ convex $\implies f(S)$ convex

Trying to prove this since no optimization book in my hands proves it. The problem is that I know nothing about $f$. Here is my pathetic attempt Since S is convex, then $tx + (1 - t)y \in S$ for $t ...
1
vote
2answers
220 views

Prove Four Statements Are Equivalent

I have the following problem, where $G$ is a graph with $n$ vertices, prove the following statements are equivalent: 1. $G$ is connected and acyclic 2. $G$ is connected and has $n-1$ edges 3. $G$ ...
1
vote
1answer
60 views

Monotone covergent sequence

Let $c>0,a_1=c/2,a_{n+1}=\frac{1}{2}(c+a_n^2)$. Determine $c$ for which the sequence converges, for each $c$, and find $\lim_{n\to\infty} a_n$. I have the feeling that I should use induction but I ...