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.

learn more… | top users | synonyms

1
vote
0answers
19 views

Is there a standard notation for $(p_i-k)(p_{i-1}-k)(p_{i-2}-k)\cdots$ where $k$ is a small positive integer

For $k=0$, there is: $p_i\# = (p_i)(p_{i-1})(p_{i-2})\cdots(5)(3)(2)$ For $k=1$, there is: $\varphi(p_i\#) = (p_i-1)(p_{i-1}-1)(p_{i-2}-1)\cdots(5-1)(3-1)(2-1)$ Is there any other notation that ...
0
votes
2answers
51 views

Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
2
votes
0answers
24 views

Proof: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$

Prove: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$ This is my work so far: Let P be any point of the plane and set: $P'=T_{AB} (P)$ We want to show ...
0
votes
0answers
14 views

Upper bounds and Lower bounds (Relations Proof Problem)

So I've only recently started studying proofs and I've been using Velleman's "How to Prove it" This is a theorem from the book. I'm having a hard time on proving it. Suppose A is a ...
4
votes
5answers
273 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
0
votes
0answers
57 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
3
votes
5answers
146 views

Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$

The problem i have is: Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ Ive been trying to use different examples of similar problems like at: ...
4
votes
2answers
107 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
0
votes
0answers
44 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...
2
votes
4answers
187 views

Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
1
vote
1answer
32 views

Converse of Borel-Lebesgue in $\mathbb R^n$

Question: If every open cover of a set $X \subset \mathbb R^n$ admits a finite subcover, then $X$ is compact. Note: Definition: $X$ is said to be a compact set is if $X$ is bounded and closed. ...
0
votes
1answer
28 views

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of $\mathbb{R}$.

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of R. I know that I need to show that the complement of this set is open in order to show that this set is closed. The ...
0
votes
2answers
54 views

Prove or disprove: There exists a prime p > 3 such that p + 2 and p + 4 are also prime

I'm having a lot of difficulties with this proof. Can someone please solve it and explain to me what's going on at each step? Thank you!
1
vote
2answers
35 views

Proof by Elements to Show $D^{c} ⊆ A^{c}$

Use proof by elements to verify that for all nonempty sets $A$, $B$, and $D$ if $A ⊆ B$, $D^{c} ⊆ B^{c}$, then $D^{c} ⊆ A^{c}$. Here's the proof I have written so far. I have gotten feedback that ...
1
vote
4answers
80 views

Suppose that $F'(x)\leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x)\leq G(x)$ for all $x \in \mathbb{R}$.

Suppose that $F'(x) \leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x) \leq G(x)$ for all $x \in \mathbb{R}$. Prove or disprove. I came up with counterexample that $\cos(x) \leq \cos(x)+1 $ for all ...
1
vote
4answers
80 views

Prove that if $3|mn$, then $3|m$ or $3|n$

I am trying to prove this for integers $m$ and $n$. I tried to reach prove that $3|m$ by assuming that 3 does not divide $n$, but this is such a basic assumption of mine already that it is hard for ...
0
votes
2answers
29 views

Proving that the set of languages over an alphabet Σ is a monoid regarding concatenation

I'm practicing proofs and would like to prove that the set of languages over an alphabet $\Sigma$ is a monoid regarding concatenation by showing that the following statements are true: There is a ...
2
votes
2answers
253 views

how to prove that the following is not a regular language?

the language we want to disprove is : $$ L = \{ 0^i1^j| gcd(i,j)=1 \} $$ my attempt : i used the pumping lemma this way: consider the set of strings of the form $0^p1^q$ such that $n <=p$ and ...
0
votes
1answer
29 views

Let S ⊂ R 2 . Show that if cl(S) is convex, then S is convex.

Let $S ⊂ R^2$. Show that if $cl(S)$ is convex, then S is convex. This seems intuitive but, I am having trouble thinking of a proof or counter example.
1
vote
3answers
53 views

Proof that when repeatedly splitting a heap of marbles into two and writing down the product of the two heap sizes, the total is ${x \choose 2}$

Here is the problem in full: A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
0
votes
0answers
54 views

Proving Integrability of $sgn(\sin(\frac{\pi}{x}))$

I must show that for $f(x) = sgn(\sin(\frac{\pi}{x}))$ on $[0,1]$, that $f$ is Integrable. I know that a function is integrable if the Upper and Lower sums of $f$ coincide. That is, if $$U(P,f) - ...
6
votes
2answers
316 views

Proofs in undergraduate mathematics [closed]

What are the best introduction books to mathematical proofs in undergraduate mathematics? I know of "Proofs from the Book" by M. Aigner and G. Ziegler, but also need one that shoots to analysis kind ...
1
vote
3answers
46 views

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length.

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length. Prove or disprove. I got the idea that they are inverse functions and probably we can show ...
1
vote
3answers
37 views

How to prove that $G(x)=ax+b$ is a one-to-one correspondence where $a\neq0$ and $a,b\in\mathbb{R}$.

$G(x) = ax + b$ where $a$ is not equal to $0$ and $b$ are real numbers. Prove $G$ is a one-to-one correspondence. I understand that for every $a$ there is a corresponding $b$-value that does not ...
1
vote
3answers
63 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...
4
votes
1answer
101 views

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$ Let $G$ be a group and $C$ the center, i.e., for any $a \in C$ and any $x \in G$, $xa=ax$. So, ...
0
votes
2answers
51 views

How to write the below proof rigorously for : If $p$ is a boundary point of $S$, then $p$ is a lim point of $S$

How to write the below proof rigorously for : If $p$ is a boundary point of $S$, then $p$ is a lim point of $S$? We have: $p$ is a boundary point of $S$ means that $$\forall r\gt 0, \exists a \in ...
0
votes
1answer
54 views

Dedekind Cuts and Real Numbers

A Dedekind cut L is a nonempty proper subset of the rational numbers that: (1) Has no maximal element (2) for all a,b in the rational numbers a is in L and b < a implies that b is in L. If $D$ is ...
3
votes
1answer
49 views

Prove that $R$ is an equivalence on $\mathscr P(A)$. Is this correct?

Suppose $B\subseteq A$, and define a relation R on $\mathscr{P}(A)$ as follows: $$R=\{(X,Y)\in\mathscr{P}(A) \times \mathscr{P}(A)\mid(X\mathrel{\triangle} Y)\subseteq B\}$$ Prove that $R$ is an ...
1
vote
1answer
47 views

Proof by cases: Prove that if $x$ and $y$ belong to the set of real numbers, then $\max(x, y) + \min(x, y) = x + y$

Question: Let $x$ and $y$ be real numbers. Using a proof by cases, show that $$\max(x, y) + \min(x, y) = x + y.$$ So for this question, I'm not sure how you would apply proof by cases. I think that ...
0
votes
1answer
43 views

Proving intervals are equinumerous

a.) Show that (0, 1] is equinermous to the interval (0, 1) by giving an example of a bijection from (0, 1] to (0, 1). My attempt: ...
1
vote
2answers
39 views

Proving intervals are equinumerous to $\mathbb R$

Let $a$, $b$ elements of $\mathbb R$ with $a < b$. By combining the results of the past exercises and examples, show that each of the following intervals are equinumerous to the set $\mathbb R$ ...
1
vote
0answers
41 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
0
votes
1answer
39 views

Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$.

Here's the problem: Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$. Where I Am: I assume that I should induct on $n$ and come to the ...
3
votes
1answer
26 views

Prove: R∩R−1 is symmetric.

The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric." If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, ...
0
votes
1answer
34 views

Prove that there is a surjection $ f:X \to Y$ if and only if $ |Y| \le |X| $.

Here's the problem: Let $X$ and $Y$ be sets. Prove that there is a surjection $$ f:X \to Y$$ if and only if $$ |Y| \le |X| $$. My work so far: I am working on the following direction: If $ |Y| ...
1
vote
1answer
43 views

Prove that S is an equivalence relation on P(X)

Let $X = \{ 1, 2, 3, \ldots , 2015\}$ and $Y = \{ 1, 2, 3,\ldots, 271 \}$. Let S be the relation on power set (X) defined by For all A, B in $\mathcal{P} (X)$, $(A, B) \in S$ if and only if $|A \cap ...
2
votes
4answers
73 views

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $ 7\mid b$

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $7\mid b$ What I did: I found the possible remainders for $a^2$ are $0, 1, 2$ and $4$. I think I should say $r_7(a^2)+r_7(b^2)$ can't equal any ...
0
votes
0answers
24 views

combinatorial proof of Vandermonde's Identity [duplicate]

So I can not figure out the combinatorial proof for Vandermonde's Identity for the example $\sum_{i=0}^k \binom {k} {i}^2 = \binom {2k} {k}$ Any help would be appreciated. Figured it out, thanks :)
0
votes
1answer
25 views

generalised eigenspace is invariant proof

let $T \in \text{End}(V)$ where $V$ is a finite dimension vector space. We define $V_j(\lambda) = \ker ((\lambda I - T)^j))$ as the generalised eigenspace. I am trying to prove that this space of is ...
0
votes
0answers
35 views

Existence of non-coprime between an integer and an arithmetic sequence

Take two relatively prime numbers $m,n \in \mathbb{Z}$ (i.e. $gcd(m,n) = 1$) where $m \neq 1$. Show that: $$\forall a \in \mathbb{Z} \textrm{ s.t. } 0<a<n$$ $$\exists i \in \mathbb{Z^+} ...
2
votes
1answer
42 views

Every vector space has a basis

Prove that every vector space has a basis. I am going to use Zorn's lemma for this also here is a necessary definition regarding totally ordered subsets: one element will be contained in the other. ...
0
votes
2answers
22 views

Short proofs about integrability

If true, the prove it; if false, the provide a counterexample. a) If $f$ is integrable, but $g$ isn't, then $f + g$ is not integrable. True: Assume that $f + g$ is integrable, then $f$ and $g$ must ...
0
votes
1answer
50 views

Probability proof and graphs

Let $G = (V, E)$ be a graph with vertex set $V$ and edge set $E$. A subset $I$ of $V$ is called an independent set if for any two distinct vertices $u$ and $v$ in $I$, $(u, v)$ is not an edge in $E$. ...
1
vote
2answers
45 views

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$ "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss ...
0
votes
0answers
90 views

Edge and Vertex set proof using an algorithm

Disclaimer: This is a homework question, so no direct answers please. All that I'm looking for is a good springboard to get started from with this question, as it has been tearing me apart for the ...
-1
votes
5answers
85 views

Prove that the sequence $\sin\left(\frac{n\pi}{3}\right)$ diverges

I don't want to hear that since $sin$ is a periodic function, etc, then we are done. I would like to see a simple proof that make use of the definition of convergence of a sequence. I have tried to ...
1
vote
2answers
30 views

how to prove the uniqueness and existence of equations

I've the equation $e^x=5$, know it has the solution $x=\ln 5$. How to prove the existence before, and after the uniqueness of this solution?
0
votes
1answer
29 views

Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative ...
1
vote
3answers
17 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...