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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9
votes
2answers
58 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
votes
0answers
28 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
votes
4answers
41 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
35 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
vote
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
101 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
votes
0answers
16 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
votes
1answer
23 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
votes
0answers
42 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
votes
0answers
53 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
33 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
votes
4answers
39 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
218 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
0answers
12 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
vote
2answers
27 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
votes
1answer
27 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 ...
-2
votes
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
votes
2answers
34 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
vote
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
vote
5answers
82 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
vote
1answer
34 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
55 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
votes
0answers
20 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
55 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
votes
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
72 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 ...
8
votes
3answers
322 views

Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
0
votes
0answers
23 views

Geometric interpretation or solution of an induction problem

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
2
votes
5answers
124 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
2
votes
0answers
27 views

Help with this proof please

Let f be the Thomae function aka the popcorn function. Use the epsilon-delta to prove that f is continuous for all irrational numbers and discontinuous for all rational numbers.
0
votes
1answer
48 views

Prove that the sequence {an} does not converge by showing it is not Cauchy

Let an = {7 + 4/n if n is even, 8 - 1/n if n is odd}. Prove that the sequence {an} does not converge by showing that it's not a Cauchy sequence. This is what I have so far. Let $\epsilon$ > 0. For ...
0
votes
2answers
29 views

Next step to take in this proof by contradiction?

This is a problem from Discrete Mathematics and its Applications Here is my work so far It's similar to this other question I had Next step to take to reach the contradiction?. I am assuming ...
2
votes
3answers
46 views

Can someone verify my direct proof that if A is a subset of B, AU B = B?

This is a problem from Discrete Mathematics and its Applications I am trying to use a direct proof to do this problem. Here is my book's explanation/section on direct proof Here is my work so ...
0
votes
5answers
38 views

If $A \subset \mathbb{Z}$ and $\sup A$ exists, prove $\max A$ exists and $\max A = \sup A$.

If $A \subset \mathbb{Z}$ and $\sup A$ exists, prove $\max A$ exists and $\max A = \sup A$. I'm not quite sure how to prove that the maximum of the set $A$ exists. Any help is greatly appreciated! ...
-3
votes
1answer
35 views

Prove that $m^{2n}\equiv 1\pmod{2^{m+2}}$ [on hold]

Prove: $m^{2n}\equiv 1 \pmod{2^{m+2}}$ where $m$ and $n$ are natural numbers. Can we prove this by the principle of mathematical induction assuming both the constant $m$ and $n$ to be 1 in the ...
0
votes
2answers
34 views

Trouble solving this induction problem

Show that, for every $n\ge2$, $3^n >n(n-1)$. Well, I started by showing the base case ($n = 2$): $3^2 > 2$ Now, for $n+1$: $P(n)\Rightarrow P(n+1)$ $$3^{n+1} > (n+1)n$$ My ...
0
votes
1answer
31 views

what are the properties of the definite integral that are related to inequalities? [on hold]

what are the properties of the definite integral that are related to inequalities? I've been searching the internet and asking teachers regarding this seemingly implausible connection, but haven't ...
1
vote
4answers
131 views

Proving that $p_1p_2\mid n$ iff $p_1\mid n$ and $ p_2\mid n.$

Let $p_1$, $p_2$ be distinct primes. Using the Fundamental Theorem of Arithmetic prove that a natural number $n$ is divisible by $p_1p_2$ if and only if $n$ is divisible by $p_1$ and $n$ is divisible ...
1
vote
0answers
25 views

Definition of infimum and supremum in being greater than elements

Suppose you have a set, $\mathbb{N}$, the set of natural numbers.The proof by contradiction is simply that you assume. $a = \sup \mathbb{N}$ The definition of $\sup = a$ would then be that. $a = ...
1
vote
7answers
49 views

Induction proof of $1 + 6 + 11 +\cdots + (5n-4)=n(5n-3)/2$

I need help getting started with this proof. Prove using mathematical induction. $$ 1 + 6 + 11 + \cdots + (5n-4)=n(5n-3)/2 $$ $$ n=1,2,3,... $$ I know for my basis step I need to set $n=1$ but I ...
2
votes
0answers
41 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
1
vote
0answers
29 views

What function to use to show that the set of positive rational numbers is countable? [duplicate]

This is from Discrete Mathematics and its Applications Here is the definition of countable that the book uses and how to determine if two sets have the same cardinality Here is the example that ...
0
votes
2answers
43 views

If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$

Suppose that $k$, $n$, and $d$ are integers and $d$ is not $0$. Prove: If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$. You may not use the theorem stating the following: Let ...
0
votes
1answer
14 views

Proof using partitioned matricies (thinking)

I have an arbitrary diagonal matrix D and an arbitrary matrix A. I want to show that the jth column of the product AD is equal to the jth diagonal entry multiplied by the jth column of A. My idea ...
3
votes
0answers
43 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
1
vote
1answer
39 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...
2
votes
4answers
59 views

Proof By Induction Help? [on hold]

I've been working through proof by induction and i'm stuck on this question. Can somebody provide some help? $$\huge 2^n-1=\sum_{i=0}^{n-1}2^i\text{ for }n\ge 1$$
1
vote
2answers
46 views

If $A$ has a maximum, prove that it only has one.

Let $A\subseteq \mathbb{R}$. We say that a real number $M\in\mathbb{R}$ is a maximum of $A$ if $M$ is an upper bound for $A$ and $M\in A$. If $A$ has a maximum, prove that it only has one; and prove ...
3
votes
5answers
98 views

How to prove this limit: $\lim\limits_{n \to \infty} \frac1{\sqrt[n] n}=1 $? [duplicate]

This is what I am trying to prove: $$\lim_{n \rightarrow \infty} \frac1{\sqrt[n] n}=1 $$ How do I go about this?