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
0answers
25 views

A question on a matrix identity

Sorry for the not very specific title. I was hoping I could get some help with a result I do not understand. The following is from a book I am reading. What I do not understand is how from 9.9.6 one ...
2
votes
1answer
23 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
3
votes
2answers
73 views

Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, ...
4
votes
0answers
31 views

Number of lucky primes

The "lucky numbers" can be constructed with this sieve. The red ones are the lucky numbers. As you can see, some are prime. Is the number of lucky primes infinite? Edit: Apparently this is an open ...
4
votes
1answer
31 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
2
votes
1answer
22 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
2
votes
0answers
18 views

Real-Definite Version of the Axiom of Induction?

I've seen in the answers to a few different questions here on the Mathematics Stack Exchange that one can clearly do mathematical induction over the set $\mathbb{R}$ of all real numbers. I am, ...
4
votes
1answer
29 views

Using the Uniform Cauchy Criterion theorem

Suppose $(f_n)$ is defined, continuous on $[a,b]$, and differentiable on this open interval. Then $c \in [a,b]$ and $(f_n(c))$ converge $(f'_n)$ and converge uniformly on $(a,b)$ respectively. How ...
5
votes
1answer
41 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
1
vote
2answers
16 views

Proving $\sum_{r=1}^n(6r-2)=n(3n+1)$ by induction

A series is defined by $\sum\limits_{r=1}^n(6r-2)$. Use the method of induction to prove that $S_n=n(3n+1)$. I am at the induction step but I am struggling to rearrange $k(3k+1)+6(k+1)-2$ into the ...
3
votes
2answers
52 views

Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
1
vote
1answer
32 views

Nowhere dense subset of unit square of measure $1$

How can one construct or prove the existence of an example of a nowhere dense set in $[0,1]^2$ with Lebesgue measure $1$ using real analysis? Osgood gives an example in ...
1
vote
1answer
29 views

Determining if $\int f_n\to 0$ implies that $f_n\to 0$ in measure and $f_n(x)\to 0$ a.e.

If $f_n$ is a sequence of measurable functions on $(X,\mu)$ into $[0,1]$ and $\int f_n\to 0$, I am trying to prove (or disprove the following): (i) $f_n$ converges to $0$ in measure. (ii) For almost ...
1
vote
2answers
33 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
0
votes
4answers
42 views

Proving a recursive sequence is bounded

I'm proving that the limit of the following recursive sequence is $\dfrac{10}{9}$: $$s_0=1,\,s_n=s_{n-1}+\frac{1}{10^n}\quad\text{for }n\ge1$$ Showing that the sequence is monotonic was easy enough, ...
3
votes
4answers
78 views

Proving that $6^{2n+1} + 1$ is divisible by $7$ for $n\geq 1$ by induction

How should I go about solving a problem like this using induction? Would I: First test $(n = 1)$ so that $6^{2(1)+1} + 1 = 6^3 + 1 = 217/7 = 31$. Then assume $(n = k)$ so that you have $6^{2(k) + 1} ...
2
votes
1answer
58 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
0
votes
1answer
34 views

Proving relation for square root of complex number

How do I represent $\sqrt{1 + ja}$? I'm trying to show that it's approximately equal to $\pm(1 + \frac{ja}{2})$ when $a \leq 1$.
1
vote
2answers
23 views

Proving the well ordering principle

THe well ordering principle has that every subset of $\mathbb{Z}^+_0$ has a least element. or if $S$ is a non-empty subset of $\mathbb{Z}^+_0$ and $S = \{a_1, a_2, a_3 ... a_n\}$, then there is a ...
0
votes
0answers
30 views

Show that $\lnot\exists x\in A(P(x))$ is equivalent to $\forall x\in A(\lnot P(x))$

This exercise is from "How to Prove it" by Daniel J Velleman. Chapter 2.2 #5 Show that $\lnot\exists x\in A(P(x))$ is equivalent to $\forall x\in A(\lnot P(x))$
0
votes
1answer
44 views

Contradiction proofs

I'm a first year Physics student and I have some trouble approaching Proofs by Contradiction in some of my Math classes. Once I get the first 2 or 3 statements I can finish the proof but a lot of the ...
2
votes
1answer
36 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
1
vote
1answer
58 views

Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$

To find the characteristic polynomial of the matrix $A^2$, would I just compute $$(\lambda^2+4\lambda-5)^2 ?$$
2
votes
1answer
30 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
0
votes
1answer
16 views

For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
-2
votes
4answers
33 views

Proof $log_{r} a = log_r s \cdot log_s a $ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
0
votes
1answer
55 views

Mathematical Induction. Horses made me question my understanding

I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ...
6
votes
0answers
80 views
+50

Does there exists a positive $t$ that satisfy this given condition?

I am curious about the validity of my claim concerning the equations: $(2k-1)t+1$ (1) $(2k^2-2k)t+(2k-1)$ (2) where $k=2,3,4,...$ My claim is for almost all $k$ or for infinitely many $k$, there ...
0
votes
1answer
24 views

$A$ is positive definite if and only if $Q$ is invertible for every choice of $Q$

Note that if $A \in M_{n \times n}$, $A^{\prime}$ denotes the transpose of $A$. I proved the following theorem already: $A$ is nonnegative definite if and only if there exists a square matrix ...
4
votes
4answers
62 views

Proving that $\varphi(n)=n\prod (1-1/p)$ without using multiplicativity

$$\varphi(n)=n\prod_{p \ \text{prime}} (1-1/p)$$ Can this useful formula be derived without using the fact that Euler's totient function is multiplicative?
2
votes
5answers
51 views

$A\backslash (B\cap C) = (A\backslash B)\cup (A\backslash C)$; only one inclusion seems to work

I encountered the following problem: $$A\backslash (B \cap C) = (A\backslash B)\cup(A\backslash C).$$ So I need to prove two things: $A\backslash (B\cap C) \subseteq (A\backslash B) \cup ...
0
votes
1answer
31 views

Map between two metric spaces and their limit

I have to proof the following: Let $f: V \to W$ be a map between two metric spaces. Proof that $f(a)$ with $a \in V$ is continuous if and only if, for every converging sequence $x_n$ in V with limit ...
0
votes
1answer
17 views

Finding the upper boundary of a sequence to find it's limit

I have to determine if the following sequences converge and if they converge I have to determine to what they converge and proof this. $$ a_n = 2^{-n} \\ b_n = \frac{n^2}{n^3 -10} \\ c_n = 1 ...
0
votes
1answer
27 views

Could the Hamel basis of $\mathbb{R^Z}$ be the set $\mathbb{R^Z}-{\mathbf{\{0\}}}$?

This is the follow up question to this question (*) According to page 2 of this link 1 and this link 2, $\mathbb{R^Z}$ (which is referred as $\mathbb{R^\infty}$ in link 1) has elements of the ...
2
votes
1answer
33 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
0
votes
2answers
15 views

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Starting off the proof of $C(X) \cup C(X)^{\perp} = \mathbb{R}^n$

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Prove or disprove the following statement: Every vector in $\mathbb{R}^n$ is in either $C(X)$ or $C(X)^{\perp}$ or both. I interpret ...
4
votes
0answers
18 views

Why is this a corollary of this theorem?

Lang - Algebra p.251 Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $Aut(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over ...
2
votes
4answers
69 views

Is there a book on proofs with solutions?

I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my ...
2
votes
2answers
55 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
13
votes
7answers
166 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
2
votes
1answer
45 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
0
votes
1answer
27 views

The Change-making problem algorithm proof (at the dynamic programming method)

I saw here the algorithm for the "Change-making problem" (at the dynamic programming method). I saw it here: http://www.columbia.edu/~cs2035/courses/csor4231.F07/dynamic.pdf I'm trying to find a ...
0
votes
0answers
24 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
2
votes
1answer
37 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
4
votes
1answer
43 views

Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
1
vote
2answers
35 views

Leibniz Formula, proof of alternating property

$$F_{A} := \sum_{\sigma\in S_n}\operatorname{sign}(\sigma) \prod_{i=1}^n A_{i \sigma(i)}$$ I am trying the prove that $\det(A)=F(A)$. I know that to do this, I need to show that $F$ satisfies the ...
0
votes
1answer
29 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
1
vote
1answer
23 views

Proving that the ball is converx

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?
1
vote
1answer
36 views

Help to understand this proof in Fulton's book

I'm reading Fulton's algebraic curves book on page 105 and I didn't understand this proof: 1.Why if $R=k[X_1,\ldots,X_n]$, then $\Omega_k(R)$ is generated (as R-módule) by the differentials ...
0
votes
1answer
63 views

Proving set identities

I am attempting to work on some proofs for my math assignment, but I'll be honest in that I am really struggling to understand them. I read through the power point given by my teacher; however, even ...