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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0
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1answer
35 views

Covering Class and Describing Outer MEasure for General Measures

I am uncertain if my description is correct, but I describe the measure in a piecewise type fashion. In general, $\mu_{\lambda}^*(A) = \infty$, if $A = X$ or $A$ uncountable. $\mu_{\lambda}^*(A) = ...
1
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2answers
45 views

Proof by induction for $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ for $k > 4$

I was given this proof for hw. Prove that $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ So, far I've gotten this Basis: $k = 5$, $2^{5 + 1} - 1 > 2\cdot5^2 + 2\cdot5 + 1$ => $63 > 61$ (So, the basis ...
4
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1answer
906 views

Prove that Square Root of a prime number is Irrational through contradiction.

I know that this question has been asked but I just want to make sure that my method is clear. My method is as follows: Let us assume that the square root of the prime number is rational. Hence we ...
0
votes
1answer
31 views

If $a\pmod 3 \equiv 1$ and $b\pmod 3 \equiv 2$, then $ab \pmod 3 \equiv 2$

I'm stuck on this this problem: Let $a$ and $b$ be positive integers with $a\pmod 3 \equiv 1$ and $b\pmod 3 \equiv 2$. Prove that $ab \pmod 3 \equiv 2$. I think the first step for the direct ...
0
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1answer
101 views

Prove that all subsequential limits are contained within a closed interval

Let $a, b$ be two real numbers such that $a < b$, and suppose that $(s_n)_{n=1}^\infty$ is a sequence such that $\forall\,\, n\,\, a \leq s_n \leq b$. Prove that all subsequential limits are ...
0
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2answers
22 views

Vector proof that $d_1^2 + d_2^2 = 2a^2 + 2b^2$ in a parallelogram

How would one prove the equality of the sum of squares of diagonals and twice the sum of squares of the two sides: $$\left|\mathbf{p} + \mathbf{q}\right|^2 + \left|\mathbf{p} - \mathbf{q}\right|^2 = ...
1
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0answers
82 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
0
votes
1answer
28 views

Euclidean algorithm to provde gcd's and multiples

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b). I was going to try setting it up, by literally doing: nb = rna + k and so forth, but something tells me this ...
1
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1answer
54 views

Tangent space and implicit function theorem

Let's say we have a $C^1$-function $f:X\to\mathbb{R}^m$ ($X\subset\mathbb{R}^{n+m}$ an open set) and the rank of the matrix $Df(x)$ is $m.$ We'll let $Z=\lbrace x\in X:f(x)=0\rbrace$ and take some ...
-1
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5answers
81 views

Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
7
votes
0answers
223 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
1
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3answers
46 views

Summation Proof

I'm getting stuck halfway through this: Show that $$\sum_{i=1}^n (y_i - \bar y_s)^2 = \sum_{i=1}^n (y_i)^2 - n\bar y_s^2$$ My skills with manipulating sums are quite rusty. I multiply the left side ...
1
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1answer
56 views

Prove with epsilon delta the limit of $5x^3$

I try to prove the $\lim_{x \to a}5x^3$ with the epsilon-delta theorem for every real a. I already came up with the idea of $0<|x-a|<\delta$ Since $|5x^3 - 5a^3| = 5|x-a||(x-a)^2 +3ax|$ Let ...
0
votes
4answers
102 views

Is what I've done a proof? Proving there is always an rational number between two distinct rational numbers

The exercise I am working on is about proving whether there is always a rational number between two other distinct rational numbers. I came up with this $\frac{a}{b} < \frac{ad + bc}{2bd} < ...
1
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1answer
111 views

Proof for the number of perfect matchings in complete graph.

I'm working on a question: Let $P_n$ be the number of perfect matchings in $K_{2n}$. Prove by mathematical induction that for each integer $n\geq1$, $P_n$ is the product of odd integers from $1$ to ...
2
votes
1answer
54 views

Continuously differentiable functions on open convex set in $\mathbf{R}^n$

This is related to a few problems I was given in class, so please try not to post full answers, and hints/methods of proof instead. I have been told that if we are given an open subset ...
1
vote
1answer
22 views

$C^1$ function on a convex subset of $\mathbf{R}^n$

I am working on the following problem given in class. Say we have a $C^1$-function $\varphi:X\to\mathbf{R}^n,$ where $X\subset\mathbf{R}^n$ is a convex set, ie. $a+\lambda(b-a)\in X$ for all $a,b\in ...
1
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2answers
39 views

Proof for $|1 - z| \geq 1 - |z|$ for $|z| < 1$, $z \in \mathbb{C}$

I can prove it "by picture" by drawing a picture of a circle of radius $|z|$ centered at $(0, 1)$. Then $1 - |z|$ is the length from the origin to the intersection of the circle with the x-axis (to ...
0
votes
3answers
47 views

Use the Fundamental Theorem of Arithmetic to prove that if a>1 is composite, then there exists a prime p such that p|a and p≤√a

I know that since $a>1$ is composite, then it can be broken down into a product of prime factors, by Fundamental Theorem of Arithmetic. So then $a=p_1p_2\dots p_k$ for some natural number k. Then, ...
1
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2answers
81 views

Is everything right in this set-theory problem?

I've got a following homework to solve: $f:\Bbb{N}^\Bbb{N}\rightarrow \mathcal P(\Bbb{N})$ is such a function that $f(\phi)=\phi(\Bbb{N}). $ Is $f$ bijective? Find $f^{-1}(B)$ where $B$ is a set of ...
0
votes
4answers
70 views

Cardinality of sets of functions

Show that the set $A$ of all functions $f:\mathbb{Z}^{+} \to \mathbb{Z}^{+}$ and $B$ of all functions $f:\mathbb{Z}^{+} \to \{0,1\}$ have the same cardinality. I am having trouble to define a ...
1
vote
1answer
44 views

Showing a function attains its maximum (proof strategy)

This is a question for a class, so please try to avoid posting full answers. I'd like to ask about the strategy of proof for showing that the mapping $\mathbf{R}^n\to\mathbf{R}$ given by ...
1
vote
2answers
37 views

Show that the intersections of the $G_s$ is normal subgroup of $G$

I need to prove that given a group $G$ acting in a set $S$, the intersection of the stabilizers $G_s$, where $G_s:=\{g\in G: g.s=s\}$ and $s$ varies through all $S$, is a normal subgroup of $G$. But ...
0
votes
1answer
118 views

Limit inferior and limit superior proof

Show that if $f$ is bounded on $[a,x_0)$, then $\varliminf_{x\rightarrow x_0^-} f(x) \le \varlimsup_{x\rightarrow x_0^-} f(x)$. I have absolutely no idea how to go about this. I can't understand the ...
0
votes
0answers
89 views

Real Analysis: Show that g is integrable on [a,b] and that $\int_a^b$ $g(x)dx=$ $\int_a^b$ $f(x)dx$

Suppose f is integrable and g is bounded on [a,b], and g differs from f only at points in a set H with the following property: For each $\epsilon>0$, H can be covered by a finite number of closed ...
3
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4answers
56 views

Proving that $\limsup_{n\to\infty}\frac{1}{n}\sum_{m=1}^n s_m\leq \limsup_{n\to\infty}s_n.$

I am reviewing for my first year analysis exam and am stuck on a problem. Let $\sigma_n=\frac{1}{n}\sum_{m=1}^n s_m$. I am trying to show that, if $(s_n)$ is a bounded sequence of real numbers, ...
1
vote
4answers
129 views

$\gcd(p, (p-1)!) = 1$?

Let $p$ be a prime number. Prove that $\gcd(p, (p-1)!) = 1$. I've attempted using the definition of $\gcd$ to solve this, but I haven't reached a conclusion. Any ideas?
2
votes
1answer
65 views

$a^n\mid b^n$ if and only if $a\mid b$.

Suppose $a$, $b$, $n$ are positive. Prove that $a^n\mid b^n$ if and only if $a\mid b$. I know that this can be proved through prime factorization, but I want to prove it using other methods. I ...
0
votes
0answers
27 views

Probability Integral Transform of Discrete RV - Equating CDF's

If $F_X$ is the cdf of a discrete random variable X (with support of all integers) and $Y \sim $Uniform(0,1) such that $F_X^{-1}(y) = \mathrm{inf} \{x:F_X(x) \ge y\}$ (allow probability integral ...
2
votes
1answer
68 views

Showing a set is a subset of another set

I need to show that $(A \cup B) \subseteq (A \cup B \cup C)$ My Work So Far: What I really need to show is that $x \in (A \cup B)$ implies $x \in (A \cup B \cup C)$ So I translated my sets into ...
0
votes
2answers
86 views

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b).

For this problem, would I be able to say that by the properties of divisibility, if the GCD divides a and b, then it should also be able to divide any multiple n of a and b?
1
vote
1answer
29 views

$\sum_{n=1}^{[a]} a_n f(n) = - \int_1 ^{a}A(x)f'(x)dx + A(a)f(a) $

Let $\{a_n\}$ be a sequence of real numbers. For $x \geq 0 $, define : $A(x) = \sum_{n=1}^{[x]} a_n$ where $[x]$ refers to the greatest integer function Let $f$ have a continuous derivative in the ...
0
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1answer
40 views

Prove that $M_{n\times n}(K)$ and $P_{n^2-1}[x]$ are not isomorphic rings

Prove that $M_{n\times n}(K)$ and $P_{n^2-1}[x]$ (polynomials with degree less than or equal to $n^2-1$) are not isomorphic rings for any field $K$ and $n\ge 2$ Let $f: M_{n\times n}(K)\to ...
1
vote
4answers
304 views

Prove that if both $ab$ and $a + b$ are even, then both $ a$ and $b$ are even.

Let $a$ and $b$ be integers. Prove that if both $ab$ and $a + b$ are even then both $a$ and $b$ are even. I've seen some solutions but they're not worded in a very simple way. Any help would ...
0
votes
0answers
225 views

Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
4
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1answer
275 views

Proof by counter example of optimal solution for Coin Changing problem (no nickels)

I'm a tutoring a student whose working on the classical coin changing problem. For those who are unfamiliar with problem or the greedy algorithm used for it. The goal is find the fewest number coins ...
1
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1answer
74 views

Find all equivalence classes

Let R by a relation defined on pairs $(m,n)$ of integers $m$ and natural numbers $n$ by $(i,j) R (k,l)$ if $il=jk$. Prove that this is an equivalence relation and give the equivalence cases. Show ...
3
votes
2answers
72 views

$P(AB=BA)$ , $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$

Let $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$ ($p$ a prime number). Find the probability $P$ that $AB=BA$ that is $P(AB=BA)$ $$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} ...
2
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3answers
87 views

Prove that if A is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. (how to when A is singular?)

Prove that if $A$ is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. first of all, since I did it about 3 or 4 times before, I started off by proving the case where $A$ is an INVERTIBLE ...
0
votes
2answers
38 views

Find integers $r$, $s$, and $t$ such that $12r + 30s + 18t = 2$

Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
4
votes
2answers
174 views

Walk on Earth: Math Puzzle

Here's the famous math puzzle posted by Prof. Walter Lewin about a person walking on earth, quoted below for posterity: A person stands on the North Pole. She walks 10 miles South, then 10 ...
2
votes
2answers
127 views

Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.

Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries. I'm really thrown off by this one, its unlike all the examples I've seen.. ...
0
votes
1answer
39 views

Return to sum of powers question.

Previously I had asked a question about a Diophantine equation linked here. I have come back to think about this question but in a different manner. So here is the set up: Let A and $a_i$ be ...
1
vote
3answers
72 views

Proof by induction of a sum?

Let $n ∈ N$. Prove by induction that there are $n$ ways to write the number $n$ as a sum $n=x_1+x_2+...+x_k$ where the $x_i$ are natural numbers and $x_1 ≤x_2 ≤...≤x_k ≤x_1+1$. For example, $5 = 5$, ...
0
votes
1answer
63 views

Real Analysis Question: derivatives

Let $$f''(x)+p(x)\cdot f(x)=0$$ and $$g''(x)+p(x)\cdot g(x)=0$$ where $a<x<b$. 1 ) Show that $W=f'g-fg'$ is a constant on $(a,b)$. 2 ) Prove: If W$\neq$0 and $f(x_1)=f(x_2)=0$ where $a \lt x_1 ...
2
votes
1answer
53 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$ [closed]

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
7
votes
2answers
152 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
1
vote
4answers
418 views

Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ

I can do it by induction, thanks to the wonderful people of this website, but I'm not sure how to do it by the Division algorithm. Can anyone help me? I think I can show how 3 divides 2n, but I'm not ...
2
votes
2answers
55 views

Show that $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$ is continuous, with $f$ continuous.

The entire problem statement is, Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Define $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$. Show that $\Gamma_f$ is continuous. My attempt ...
0
votes
3answers
102 views

Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$

Would this be the same thing as saying "Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for any $s, t \in \mathbb{Z}$"? I can do the proof for any integers $s$ and $t$, but if any and all mean the same ...