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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1answer
26 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
1
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1answer
35 views

For how many integers is this a prime number?

For how many integers $n$ is: $9 - (n-2)^2$ a prime number? I want to try this using a rigorous definition of prime number/ actual problem rather than try-error? Please only give hints, so I can do ...
0
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1answer
34 views

Proving an “OR” statement

If one wants to proof $P\vee Q$, is it sufficient to proof $\lnot P \rightarrow Q$? Because it makes intuitively more sense to me that $P\vee Q$ would be logically equivalent with $(\lnot P ...
0
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1answer
34 views

For what natural number $n$ is the following inequality true: $2^n \geq 2\cdot n^2$?

Can you solve this by using induction? The inequality is true for $n = 1$, but is false until $n = 7$. After the induction step I got $$2^n \geq n^2 + 2n + 1.$$ If you take the limit as $n$ ...
0
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1answer
19 views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
-1
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1answer
17 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
0
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2answers
29 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
0
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1answer
56 views

What is the contraposive of this statement?

I have to prove the negation of this statement: $$\forall a,b,c\in\mathbb{Z}{\;if\;a\;|\;b\} $$ But the fact that there is a "and" is very disturbing. I think that I am missing something because my ...
3
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3answers
53 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
0
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2answers
43 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
1
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1answer
21 views

Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional

I am having trouble understanding this whole question, and how to prove it. Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of ...
1
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1answer
42 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
0
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1answer
25 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
1
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2answers
99 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
-1
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1answer
16 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
0
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1answer
44 views

proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let $n ≥ 0$ and $k ≥ 0$ be integers. 1) How many bitstrings of length $n + 1$ have exactly $k + 1$ many $1$s? 2) Let $i$ be an integer with $k ≤ i ≤ n$. What is the number of bitstrings of length $n ...
-1
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2answers
26 views

Prove that if $y>1$, then $\forall M\in\mathbb{R}$, there exists an $N$ in the natural numbers s.t. $n\geq N$ implies $y^n>M$. [on hold]

For $y\in\mathbb{R}$, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N\in\mathbb{N}$ such that $$ n≥N \implies y^n>M. $$ I'm not used to proving these kinds of questions so any ...
3
votes
1answer
59 views

Could someone take a crack at this number theory problem?

The question is stated as follows: If $\mathrm{gcd}(a,m)=1$ and $X$ is a complete residue system $\bmod m$, then the set obtained by multiplying each member of $X$ by $a$ is also a complete residue ...
1
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1answer
28 views

Define $x^3$ = $x \times x^2$. Prove that if $x_1, x_2$, . . . represents $x$, then $x_1^3$, $x_2^3$, . . . represents $x^3$ [on hold]

I'm a little bit lost on where to start this problem. My initial thought is to work backwards. Say $x^3$ is a Cauchy sequence. Then for some $j, k \geq m$ contingent on $n$, we have |${x^3}_j - ...
0
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1answer
27 views

Prove the associative law for the addition of real numbers

The problem asks us to prove the commutative and associative laws for the addition of real numbers. The commutative proof seems straightforward. I am wondering how to approach the proof of the ...
0
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0answers
19 views

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
3
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1answer
55 views

Relationship between increasing integer sequences

Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal ...
10
votes
3answers
84 views

Prove that if $\left({x+\sqrt{x^2+1}}\right)\left({y+\sqrt{y^2+1}}\right)=1$ then $x+y=0$

Let $$\left({x+\sqrt{x^2+1}}\right)\left({y+\sqrt{y^2+1}}\right)=1$$ Prove that $x+y=0$. This is my solution: Let $$a=x+\sqrt{x^2+1}$$ and $$b=y+\sqrt{y^2+1}$$ Then $x=\dfrac{a^2-1}{2a}$ and ...
4
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0answers
30 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
0
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4answers
49 views

Prove that a continuous real function with finite limits is bounded

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Assume that $\lim_{x \rightarrow \pm \infty} f(x)$ exist and are finite. Prove that $f$ is bounded. So to show that $f$ is bounded, I ...
0
votes
0answers
62 views

Prove the limit as $x$ approaches $0$, $\frac{\sin(x)}{x}$ approaches $1$ using the epsilon delta definition [duplicate]

Prove that $\lim_{x\to0}\frac{\sin(x)}{x} = 1$. So far i have things such as $|\sin(x)|\leq|x|$ for small $x$ and $|\sin(x)|\leq1$ so it is bounded but I'm rather stuck, Also I am not looking for a ...
2
votes
1answer
39 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
1
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0answers
15 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
9
votes
3answers
109 views

Prove that $a+b$ cant divide $a^a+b^b$ nor $a^b+b^a$

Let a and b be natural numbers so that $2a-1,2b-1$ and $a+b$ are prime numbers. Prove that $a+b$ cant divide $a^a+b^b$ nor $a^b+b^a$. I get that $gcd(a,b)=1$. I havent got anything special for now ...
0
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0answers
17 views

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$ (or $n-1$ and proving for $n$)? both with induction. The first one is ...
0
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1answer
27 views

Help with Conic: Hyperbola's chord of contact

please help with this proof. "Show that the tangents at the endpoints of a focal chord of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ meet on the corresponding directrix." This is a ...
0
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0answers
44 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
0
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0answers
57 views

On correctness of induction proof

I want to prove a certain property $\mathsf{P}$ on every multiaffine polynomial in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$. Supposing I show property $\mathsf{P}$ to be valid at $n\geq9$ variable ...
3
votes
2answers
36 views

Is my deduction of $t$ being true logically correct?

According to the problem on my homework (yes, this is my homework), number 42 in chapter 2.3 of Discrete Mathematics with Applications by Susanna S. Epp, the following are true: \begin{align} ...
0
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4answers
41 views

I'm having trouble understanding a step of induction.

The problem my teacher presented was to prove, $(1 + x)^n \geq 1 + nx$ for all real numbers $x > -1$ and integers $n \geq 2$. The way it was done in class is: $(1+nx)(1+x) ≤ (1+x)^n (1+x) $ ...
2
votes
2answers
221 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
0
votes
1answer
20 views

Use the least integer principle to prove the following.

Least integer principle: Every non-empty set of positive integers has a least element. Using this fact, define $r$ to be the least integer for which $j - qk > 0$ where $j, k \in \Bbb{Z}$ ...
1
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2answers
28 views

Changing the state of coins and finding the minimum number of steps to do it

I have $N$ coins all showing heads. At each turn, I change the state (i.e., a head is changed to a tail, vice versa) of $N-1$ coins. Prove that all the coins can end up showing tails if and only if ...
0
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1answer
28 views

Rigorous proof for a maximization problem

Problem: Eight players entered a round-robin tennis tournament. At the end of the tournament, a player who wins $N$ sets will take home $N^2$ dollars. The entry fee is $17.50 per player. Why is this ...
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0answers
30 views

Proof: A+B is upper triangular [on hold]

Assume A and B are nxn matrices. Prove that A+B is upper triangular.
2
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2answers
36 views

Is $\sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous?

$f(x) = \sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous because: Proof: $f(x)$ is a continuous function on $[0, 1]$, which is a closed interval, so $f$ is uniformly continuous on ...
1
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1answer
47 views

A question about a linear algebra proof [on hold]

If $f(x)$ is a function with domain $R$ such that for all real $a, x$ it is $f(ax) = af(x)$ then there exists a real number $b$ such that $f(x) = bx$ for all $x.$ How to prove this statement?
1
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5answers
95 views

how to prove: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ [duplicate]

need help to prove this: $\sum\limits_{k=1}^n k\binom{n}{k}=n \cdot 2^{n-1} $ where $n$ is integer $\geq 1$. Question also said taking the derivative of $(1 + x)^n$ would be helpful which I've found ...
1
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1answer
35 views

Clarification on Cantor Diagonalization argument?

My book is Discrete Mathematics and its Applications. This is its section on Cantor's Diagonalization argument I understand the beginning of the method. The author is using a proof by ...
2
votes
3answers
56 views

How to approach this proof problem, what proof to use, what assumption to use?

This is a problem from Discrete Mathematics and its Applications Here is the definition of rational that my book uses Usually when I approach this type of a problem, I can find a type of proof to ...
0
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0answers
22 views

Would it be necessary to have another proof within the proof by cases in this problem?

This is a problem from Discrete Mathematics and its Applications I am using Proof by Cases. This is my book's definition on it. Here is my work so far I tried to leverage without of generality ...
3
votes
5answers
57 views

Manually obtainining a list of primes $\leq n$, by using the root of n?

In my abstract math class I learned that if we want to get a list of primes $\leq n$ manually, we have to calculate the root of n, and the floor of that result will be the greatest number for which to ...
0
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0answers
30 views

Solving a proof by combinatoric method

Any good questions you guys have in mind?: prove the following equation by coming up with a combinatoric problem and solving it step by step (Solve combinatoric method): $$ {n \choose 1} + 14{n ...
2
votes
1answer
74 views

How many ordered triples $(a, b, c)$ exist?

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Breaking it up, $500 = 2^2\cdot5^3$ $abc = 2^2 \cdot 5^3 = 2\cdot 2 \cdot 5 \cdot 5 \cdot ...