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.

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2
votes
1answer
70 views

Uniform Convergence of $nx^{2}(1-x)^{n}$ on $[0,1]$

Uniform Convergence of $nx^{2}(1-x)^{n}$ on $[0,1]$ My attempt: criterion: suppose $f_n:I\to\ J$ is a sequence of functions which converges point wise to a function $f$, then the convergence is ...
6
votes
2answers
441 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
2
votes
2answers
82 views

Prove $A-I $ is not invertible. [closed]

If A=($a_{ij}$) is a nxn matrix of real numbers such that $\sum_j^n$ $a_{ij}$=1 for each i, show that the matrix A-I is not invertible. I am not very good at proofs and have previously heard you ...
1
vote
6answers
79 views

Prove that the limit of $a_n = \frac{2n^3+n^2}{(n+2)^3}$ is $2$.

if $a_n = \dfrac{2n^3+n^2}{(n+2)^3}$, prove that the limit of $a_n$ (as $n$ tends to infinite) is $2$ using the definition of a limit. My attempt was $\left |\dfrac{(2(n^3))+(n^2)}{(n+2)^3} - ...
0
votes
1answer
45 views

Prove the sequence $\{\frac{1}{a_n}\}\,\, (n\geq 1)$ is Cauchy.

Given that $\{a_n\}$ ($n\geq 1$) is a Cauchy sequence, prove that if there exists an $r$ greater than $0$ such that for all $n$ in the natural numbers $a_n$ is greater than $r$, then ...
1
vote
2answers
36 views

I have a question about “if and only if” propositions.

I have the following proposition: $m - n = p - q$ if and only if $m + q = n + p$ From what I understand, $A = B$ if and only if $C = D$ means two statements: if $A = B$ then $C = D$ and if $C = D$ ...
-1
votes
3answers
69 views

If $a \equiv b$ (mod 2n), then $a^2 \equiv b^2$ (mod 4n)

How would I go about proving: If $a \equiv b$ (mod 2n), then $a^2 \equiv b^2$ (mod 4n)? I already tried proving $a+b = 2nk$ for some integer k, and that was pretty straightforward. But when I try to ...
1
vote
2answers
50 views

A basic proof: $\forall a,b\in\mathbb{Z}$, if $a\left.\right|b$ then $a^2\left.\right|b^2$

I must state whether the following is true or false on my homework (yes, this is a homework problem, so I would appreciate it if you would only give hints or suggestions and not write out the total ...
2
votes
3answers
27 views

Prove that $z_n \rightarrow z_0$ if and only if $\bar{z_n} \rightarrow \bar{z_0}$ as $n \rightarrow \infty$.

In my Complex Analysis course, I'm supposed to prove that. I'm not really too sure where to start, however. Any pointers? Thanks. :-)
-1
votes
3answers
88 views

Prove that $(x_1+x_2)^2 \neq x_1^2+x_2^2$

For $x_1, x_2 \in \mathbb{R}$ there is the rule: $(x_1\times x_2)^2=x_1^2\times x_2^2$. How can I prove, that this rule doesn't apply for: $(x_1+x_2)^2$?
-1
votes
1answer
170 views

The union of two sets A, B is the set AUB. Prove that if A and B are nonempty bounded subsets of R, then AUB is bounded and supAUB = max{supA, supB}.

Proof: If A,B C R and are nonempty and bounded, ==> There exists a least upper bound M s.t: x <= M for all x in A or x in B, by the Completeness Axiom. If A and B are bounded ==> AUB is bounded. ...
0
votes
1answer
64 views

Proof of an alternate form of the triangle inequality

Since it is all positive squaring does not change anything. So: $$ (a_1^2 + \cdots + a_n^2) + 2\sqrt{(a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots b_n^2)} + (b_1^2 + \cdots + b_n^2) \ge (a_1 + b_1)^2 + ...
0
votes
1answer
22 views

Gradient of composition

Consider a function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined by $g(x) := \langle a,x \rangle = a_{1}x_{1}+ ....+ a_{n}x_{n}$, for some $a \in \mathbb{R}^{n}$. Then consider the function ...
1
vote
2answers
48 views

Proof by induction $n^2-2n-1>0$ for $n \ge 3$

I want to use induction to prove that $n^2-2n-1>0$ for $n \ge 3$ Base case: $3^2-2(3)-1>0$ $ \space \checkmark$ Inductive step: $(n+1)^2-2(n+1)-1>0$ $\iff n^2+2n+1-2n-2-1>0$ $\iff ...
0
votes
0answers
60 views

Euler's theorem of homogeneous function (I dont understand the proof)

Here is a link for those who want to take a look at the theorem. (http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf) I considered a function g(t)=f(tx) for fixed x and I took a ...
0
votes
0answers
9 views

Are these correct solutions for two prepositions?

I have two propositions that I need to prove using the addition and multiplication axioms as well as two propositions that I have proven: A) For all $m, n \in\mathbb Z, (-m)(-n) = mn$ B) For all $m ...
2
votes
0answers
23 views

Give counterexamples of some assertions in probability

I have the following exercise: An event $F$ is said to carry positive information about an event $E$, and we write $F \uparrow E$ if $P(E|F) \ge P(E)$ Prove or give counterexamples of the following ...
2
votes
2answers
59 views

Prove $\lim_{x\to \infty} \frac{4x^2 - 7}{2x^3 - 5} = 0$ using $\epsilon$-$N$ limit definition

I am having difficulties manipulating the problem so that I can find a $N$ value to choose. Suppose $x > N$, then $$\left|\frac{4x^2 - 7}{2x^3 - 5}\right| \leq \frac{4x^2}{|2x^3 - 5|} + ...
0
votes
0answers
31 views

Proving the additive inverse of a sum is the sum of the additive inverses

I have this proposition to prove: For all $m, n, \in\mathbb Z$: $-(m + n) = (-m) + (-n)$ Proof: \begin{align*} -(m + n) + (m + n) &= 0 + 0\\ -(m + n) + (m + n) &= (-m) + m + (-n) + n\\ (-m) + ...
0
votes
0answers
45 views

Proof technique question related to Euler's theorem for homogeneous function

I am trying to prove Euler's theorem for homogeneous function. Actually, proof is given http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf What I don't understand is that proof is based ...
0
votes
1answer
39 views

Events that carry negative information

I want to answer the following exercise: An event $F$ is said to carry negative information about an event $E$, and we write $F \downarrow E$ if $P(E|F) \le P(E)$ Prove or give counterexamples of ...
1
vote
2answers
35 views

I am stuck at this proposition

I am stuck at this proposition: Let $x \in\mathbb Z$. If $x \cdot x = x$, then $ x = 0$ or $x = 1$. I am learning the "or" statement and would greatly appreciate any hint. Thank you! Here are the ...
1
vote
1answer
50 views

injective functions require functions whose composition equal the identity function

I'm trying to prove: Let $A, B$ be sets and let $f : A\to B $ be a function. Prove that if $f$ is one to one then there is a function $g : B\to A$ so that $g(f)=id_A$ Here is what I have, Suppose $f ...
1
vote
0answers
20 views

Proof a different generalized derivative form [duplicate]

Prove that: $$f'(x)=\lim_{h,k\to 0^+}\frac{f(x+h)-f(x-k)}{h+k}.$$ The denominator is the issue. I thought of $u = h + k$ but that created an issue for the limit bounds. I tried adding and ...
1
vote
1answer
87 views

Use of “without loss of generality” in inequalities

I would like to ask about WLOG assumptions in inequalities. I know that in symmetric inequality we can WLOG assume $a\ge b\ge c$, in cyclic inequality we can WLOG assume $a=\max(a,b,c)$. What about ...
0
votes
0answers
23 views

Prove that arbitrary $z_1,z_2\in S$ satisfy $|z_1-z_2|\le3$

Let $S=\left\{z\in\mathbb{C}|\left|z^2+1\right|=\left|z+i\right|\right\}$. Prove that arbitrary $z_1,z_2\in S$ satisfy $|z_1-z_2|\le3$. This is my solution: First notice that $z^2+1=(z+i)(z-i)$, so ...
0
votes
0answers
12 views

Proof technique question about fixed vector and every vector.

I am trying to prove Euler's theorem about homogeneous function. One thing that I want to make sure is that if you prove something for some fixed vectors, does that automatically follow that it is ...
1
vote
1answer
38 views

Can someone verify this direct modulus proof?

This is from Discrete Mathematics and its applications To do this proof, I used this mod property Here is my work What I did was basically expand both sides of (a-c) mod m and (b - d) mod m ...
3
votes
3answers
96 views

Prove the uniqueness of subtraction

I have to prove this proposition: Given $m,n \in\mathbb Z$, there exists one and only one $x \in\mathbb Z$ such that $m + x = n$. So, just to be sure: I am given an equation and asked to first prove ...
2
votes
2answers
51 views

Show 3 propositions are equivalent by proving a given set of implications

I am giving the following propositions: $p: a < b$ $q: \frac{a+b}{2} > a$ $r: \frac{a+b}{2} < b$ $a$ and $b$ are real numbers. I need to show that these are equivalent by proving the ...
1
vote
0answers
18 views

Proof for two prepositions

I have two prepositions to prove with the basic axioms for addition and multiplication: 1) For all $m \in\mathbb Z$, $-(-m) = m$ \begin{align*} (-m) + -(-m) &= 0\\ (-m) + m &= 0 \\ (-m) + ...
0
votes
1answer
58 views

Show that if $\lim a_n = L$ and $a_n > a$ for all $n$, then $L \geq a$.

Proof: We know that $\lim a_n = L$ and $a_n > a$ for all $n$. Apply the limit to both sides: $\lim a_n > \lim a \Longrightarrow L > a$. Thus, $\lim a_n = L$ and $a_n > a$ implies that ...
0
votes
2answers
57 views

Prove directly from the definition of the limit that $\lim (1/n!) = 0$

I'm having a hard time finding my number $N$. This is what I have so far: $$\left|\frac{1}{n!}\right| < \epsilon$$ $$ \frac{1}{n!} < \epsilon \Rightarrow n! > \frac{1}{\epsilon}$$ I know ...
3
votes
2answers
86 views

Is this proof not basic enough?

I have the following proposition: For all $m \in\mathbb\ Z, m \cdot 0 = 0 = 0 \cdot m$ Here is my proof: \begin{align*} m \cdot 0 &= m \cdot (m + (-m))\\ m \cdot 0 &= (m \cdot m) + (m \cdot ...
2
votes
2answers
554 views

If $x \in\mathbb{Z}$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$

I am learning proofs, and I am stuck with this proposition: Let $x \in\mathbb{Z}$. If $x$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$. I want to use the additive ...
2
votes
4answers
185 views

proof that $\frac{d}{dx}e^x = e^x$

I am working on proving different mathematical formulas. I am currently working on proving that $\frac{d}{dx} e^x = e^x$ . This is my proof so far: $$\frac{d}{dx} e^x = \lim_{h \to0}\frac{e^{x+h} - ...
3
votes
2answers
53 views

Writing Corollaries into Proofs

I'm taking Discrete Math and one of my homework problems from Epp's Discrete Mathematics with Applications asks me to prove the following: If $r$ and $s$ are any two rational numbers, then ...
1
vote
3answers
171 views

Proving by contrapositive: x and y are integers, and xy is even, then x is even or y is even

I need to prove the following by contrapositive: "$x$ and $y$ are integers, and $xy$ is even, then $x$ is even or $y$ is even" I'm sure this question isn't very hard to solve, however my ...
2
votes
5answers
114 views

Show that $\rm lcm(a,b)=ab \iff gcd(a,b)=1$

Show that $$\rm lcm(a,b)=ab \iff \gcd(a,b)=1.$$ My attempt: If $\gcd(a,b)=1$ then there exist two integers $r$ and $s$ such that $$ar+bs=1.$$ and then I'm stuck... any advice?
-1
votes
1answer
32 views

proving that a function is homomorphism of rings

I am stuck with this problem.. Can someone please help me? I know the definition of isomorphism, homomorphism, injective, and surjective but that's pretty much it. :( so part a) it satisfies the ...
0
votes
1answer
35 views

Set and Logic, Proving two quantifier the same

$(∃x∈A:P(x))∨(∃x∈B:P(x)) = ∃x∈(A∪B):P(x)$ I spent hours approaching this problem many different way By Definition: $(∃x∈A:P(x))∨(∃x∈B:P(x)) \\ ∃x:[(x∈A \to P(x) \vee (x∈B \to P(x)] \\ ∃x:[(¬x∈A ∨ ...
1
vote
1answer
28 views

Is this a correct to prove that there are no solutions for these trigonometric functions?

This is part of a problem that I've been doing; it turns out the way I was doing it was wrong but I've still got a question about one method I used in trying to prove the problem. I ended up trying to ...
0
votes
2answers
38 views

Using the fact that 2 is prime, show that there do not exist integers p and q such that $p^2=2q^2$

Using the fact that 2 is prime, show that there do not exist integers p and q such that $p^2=2q^2$. Demonstrate that therefore $\sqrt2$ cannot be a rational number. Second attempt: Suppose ...
2
votes
2answers
52 views

Mean Value Theorem H

Let $f$ and $g$ be continuous on $[a,b]$, and differentiable on $(a,b)$ and let $f(a)=f(b)=0$. Show that there exists a $c\in (a,b)$ such that $$f'(c)=f(c)g'(c)$$ I've been stumped on this for hours. ...
0
votes
2answers
37 views

Proof of subspaces and vector spacecs

Which of the following sets W are subspaces of the given vector space V over the field F? (a) V = R^3, F = R , W = {(a, b, c) ∈ R^3|a^2 + b^2 = c^2} (b) V = m x n matrix, F = R ,W = {AB|A ∈ m x k ...
1
vote
2answers
79 views

Unique least common multiple proof

Define the least common multiple of 2 nonzero integers a and b, denoted by $\operatorname{lcm}(a,b)$ to be the nonnegatve integer $m$ such that both $a$ and $b$ divide $m$, and if $a$ and $b$ divide ...
1
vote
0answers
34 views

What is the difference between these two proofs?

I am studying proofs, and I am stuck thinking about the logic behind these two propositions: 1) Let $x \in\mathbb Z$, if $x$ has the property the for all $m \in\mathbb Z$ $mx = m$, then $x=1$. 2) Let ...
0
votes
1answer
31 views

Proofs regarding radius of convergence

Consider $\sum_{n = 0}^{\infty} a_{n}x^{n}$ with radius of convergence R. a) Prove that if all $a_{n} \in \mathbb{Z}$ and if infinitely many of them are nonzero, then $R \leq 1$ Proof: Assume $R ...
0
votes
3answers
69 views

Proving a theorem using another theorem

I need help proving this Theorem 1: If (k, b) = 1 and k|ab, then k|a. I was looking at another theorem which may help me prove the above, but it doesn't make much sense to me. Theorem 2: If a and b ...
0
votes
0answers
24 views

The cartesian product of compact sets is compact

my question this time is How to prove the following: Let $x \in \mathbb{R}^{n}$ and $E \subset \mathbb{R}^{m}$ compact, prove that $\{x\} \times E$ is compact and form here how to conclude that if ...