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
5answers
70 views

For every natural number $n$, $\gcd(an,bn)=n\gcd(a,b).$

For every natural number I am trying to show that $n$, $\gcd(an,bn)=n\gcd(a,b).$ Here is my attempt. Put $d = gcd(a,b)$; we can write $d=aT+bJ$, where $T$ adn $J$ are integers. Then as $d|a$ and ...
0
votes
1answer
22 views

Vector spaces and direct sums

The map that was constructed in lectures is: $V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$ Is it correct to generalise this to, ...
3
votes
1answer
41 views

Proof that for any $16$ digit number there is at least one sequence of $1$ or more digits which its product is a perfect square

I came across this problem where one is asked to proof that, for any $16$ digit number there is at least a sequence of $1$ or more digits which its product is a perfect square. For example, in the ...
0
votes
1answer
23 views

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

Assume that a,b,n are all natural numbers. I was going to set it up as: na = q(1)*n(b) + r(1) where a>b and go down the chain: nb = q2 * r(1) + (r2) but something seems off. Someone told me ...
0
votes
1answer
24 views

Prove that if a|c and b|c, and a and b are relatively prime, than ab|c

How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
2
votes
1answer
30 views

Proof by induction regarding maximum number of questions one can ask.

sorry for the pretty ambiguous title. It's otherwise hard to describe this problem without stating it in full. There are $n$ points drawn on a whiteboard. Between every pair of points $X$ and $Y$ ...
0
votes
3answers
32 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
1
vote
1answer
44 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
0
votes
2answers
41 views

Assume that 495 divides the integer 273x49y5 where x,y ∈ {0,1,2…9}. Find x and y.

So, I know that $495 = 5\times 9\times 11$. So then, if that's the case, then the number $\overline{273x49y5}$ must be divisible by $495$ if and only if it is divisible by 5 and 9 and 11. Then, I ...
-1
votes
0answers
25 views

Prove the continuity and differentiability of a function in a point. [duplicate]

This question Is the same the question as this one (that I have posted yesterday at 12 am that I why I disconected from a large period of time) Prove that a function is both differentiable and ...
0
votes
2answers
43 views

Let n ∈ ℕ. If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

I know how to test the divisibility of a number by 9, but only if I am given what n is. How would I set this problem up?
-2
votes
1answer
49 views

Prove $2n+3 \le 2^n$ for all integers $n \ge 4$.

I have already started the problem but I am unsure on how to proceed. Prove $2n+3 \le 2^n$ for all integers $n \ge 4$. Base Case: Choose $n = 4$. $2n + 3 \le 2^n$ $2(4) + 3 \le 2^4$ $8 + 3 \le ...
0
votes
1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
1
vote
6answers
221 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
0
votes
0answers
11 views

Proving relations are orders

The Problem Let P and Q be posets with respect to some order $\sqsubseteq$. Proof that the following relations are indeed orders. If P' is a subset of P, then P' is also a poset with ...
11
votes
2answers
123 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
1
vote
3answers
74 views

If $x^2$ is divisible by $4$ then $x$ is even?

I am studying discrete mathematics as course and I have to prove this "If $x^2$ is divisible by $4$ then $x$ is even". I am wondering how to prove it using the contrapositive of this ...
5
votes
3answers
130 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
1
vote
1answer
29 views

GCD Proof Questions?

Question 1: $a$ divides $b$ iff $d=a$, where $d=\gcd(a,b)$ My Solution: $d=at+bj$. If $a$ divides $b$ then $b=aK$. So $d=at+aKj$ hence $d=aN$. Plus as $d$ is the $\gcd$ $d=a$. And then in this method ...
3
votes
2answers
61 views

Prove that a function is both differentiable and continous at a point $x_0$

Suppose $f$ is differentialble on $(a,b)$, except possibly at $x_0 \in (a,b)$ an is continous on $[a,b]$; assume $ \lim\limits_{x\rightarrow x_0}f´(x)$ exists. Prove that $f$ is differentiable at ...
1
vote
0answers
27 views

Prove that $f$ is uniformly continous

I have to prove this: Suppose $f:(a,b)\to \mathbb{R}$ is differentiable and $|f´(x)| \leq M$ for all $x\in (a,b)$.Prove that $f$ is uniformly continous on $(a,b)$.Give an example of a function ...
0
votes
1answer
17 views

Use the mean value to prove a certain result

I need to prove the following: Use the Mean-Value Theorem to prove that: $$\sqrt{1+h}<1+\frac{1}{2}h$$ for $h>0$ My attempt: we first note that given that $h>0$ then $$1+\frac{1}{2}h ...
0
votes
2answers
31 views

Finding supremum and infimum of a given set and proving it formally

I'm trying to learn how to find supremum and infimum of a given set as this is essential in my further studying. Here's a problem I want to tackle: $A=\{\frac{n-k}{n+k}:n,k\in\Bbb{N}\}.$ Find ...
0
votes
2answers
73 views

How to determine the domain of $\ln(\sqrt{x^2-3x+2} - x)$?

I know that $$f:x\rightarrow ln(x)$$ is defined $$\forall x>\mathbb{R^{+*}}$$ But what happened when the argument of f is a function as "complicated" as$$\sqrt{x^2-3x+2} - x$$ Obviously we want ...
0
votes
0answers
19 views

Quantification Proofs [closed]

(Ɐx)Fx Premise ~Fa v Ga Premise a=b Premise Fa Line 1, UI Ga Line 2, 4, DS Gb Line 5, ? I do not know what is used to get Line 5. Can anyone please help?
0
votes
1answer
30 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) = ...
0
votes
0answers
23 views

Question on Proof (Direct, Contrapositive, Contradiction)? [closed]

Question 1: For every positive real number $x$, there is a positive real number $y$ less than $x$ with the property that for all positive real numbers $z$, $yz\ge x$. Question 2: For every positive ...
1
vote
2answers
32 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 ...
2
votes
0answers
30 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
30 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
votes
1answer
49 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
votes
2answers
19 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
vote
0answers
40 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
8 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
vote
1answer
18 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 ...
-2
votes
1answer
27 views

Need help with the proof of this statement |a + b| = |a| + |b| iff ab>= 0 [closed]

Could someone please provide the proof of the following? $$|a + b| = |a| + |b| \quad\text{iff}\quad ab \ge 0.$$
0
votes
5answers
56 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 ...
1
vote
3answers
37 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
vote
1answer
38 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
64 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
vote
2answers
34 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 ...
1
vote
1answer
19 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
17 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
vote
2answers
37 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
35 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, ...
-4
votes
0answers
29 views

Union and intersection of two pairwise disjoint families of sets [closed]

Let $\mathscr{A}$ and $\mathscr{B}$ be two pairwise disjoint families of sets. Let $\mathscr{C} = (\mathscr{A} \cap \mathscr{B})$ and $\mathscr{D} = (\mathscr{A} \cup \mathscr{B})$. (a) Prove that ...
1
vote
2answers
76 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
57 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
24 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
31 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 ...