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
25 views

Prove using a proof by contradiction: There is no smallest positive real number

Prove using a proof by contradiction: There is no smallest positive real number Let us assume the contraction: There is a smallest positive real number. How do I continue?
-1
votes
1answer
40 views

Linear dependence of two vectors [closed]

Prove or disprove that vectors $x$ and $y$ are not linearly independent $\iff \exists \alpha \in T \ni [(x = \alpha \cdot y) \vee (y=\alpha \cdot x)]$. If it's not true I can just find ...
4
votes
4answers
223 views

Mathematical Induction Question, Proof Help [duplicate]

Prove using Mathematical Induction that for all natural numbers ($n>0$): $$ \frac 1 {\sqrt{1}} + \frac 1 {\sqrt{2}} + \cdots + \frac 1 {\sqrt{n}} \ge \sqrt{n}. $$ ...
0
votes
3answers
29 views

Let n be an integer. Prove that if $2|(n^2-1)$ then $4|(n^2-1)$.

I know that $n^2=2k$ for some integer $k$. Please help me continue. i've got a midterm tomorrow that I'm really stressed about. Thanks
0
votes
5answers
54 views

Let $n $ be an integer. Prove that if $ 2|n$ and $3|n$, then $6|n$ [closed]

Let $n $ be an integer. Prove that if $ 2|n$ and $3|n$, then $6|n$ How do I prove this? I know that: $n =2k$ for some integer $k$, $n = 3q$ for some integer $q$
0
votes
0answers
57 views

Prove an implication about quadratic form definiteness

Sylvester's criterion states that a quadratic form $q$ over an $n$-dimensional real linear space $V$ is positive definite $\iff$ all main minors $\Delta_1, \Delta_2, ..., \Delta_n > 0$. In ...
1
vote
1answer
36 views

Convergence of $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$

Does $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$ converge absolutely, conditionally, or diverge? Attempt: Yes, by the ratio test we have $$ \lim_{n \to \infty} \left| \frac{(-1)^{n+1} ...
2
votes
1answer
78 views

Proof with combinatorial argument

Show with combinatorial argument that this is equal : $$\dbinom{n}{k+1} = \dbinom{n-1}{k}+ \dbinom{n-2}{k} +...+ \dbinom{k}{k}$$ I have no idea how to do that so it would be really helpful ...
0
votes
2answers
17 views

Proof on primitive elements

'Suppose x is an element of order $\phi$(n) in $\mathbb{Z}$/n$\mathbb{Z}$. Then every invertible element of $\mathbb{Z}$/n$\mathbb{Z}$ is a power of x.' The lecture taught me that when this ...
3
votes
3answers
120 views

If $f$ is continuous on $\mathbb R$ then $\exists c\in\mathbb R: f(x)=c$ has only one solution

I have to prove that there is no continuous function $f: \mathbb{R} \to \mathbb{R}$ such that, for each $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions. My attempt: We have that ...
2
votes
4answers
82 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...
1
vote
1answer
30 views

Prove the uniformly continuity of a function with a certain property

I need to prove this: Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and has the property that for each $\epsilon >0$ there is $M>0$ such that if $|x| \ge M$, then $|f(x)|< ...
0
votes
3answers
55 views

Prove that the set $A:=\{ x : 0 \le f(x) \le 1 \}$ is compact.

I have to prove the following: Suppose that $f : D \to R$ is continuous with $D$ compact. Prove that $\{ x : 0 \le f(x) \le 1 \}$ is compact. My attempt: We define $$A:=\{ x : 0 \le f(x) \le 1 ...
0
votes
0answers
39 views

Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
1
vote
2answers
101 views

Prove that there exist two integers such that i - j is divisible by n.

Here's the full question: Prove that, for any $n + 1$ integers, $\{x_0, x_1, x_2, . . . , x_n\}$, there exist two integers $x_i$ and $x_j$ with $i \neq j$ such that $x_i − x_j$ is divisible by $n$. ...
1
vote
0answers
25 views

Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a ...
1
vote
3answers
85 views

Show that there are no positive integer solutions to x^2 + x + 1 = y^2.

I'm trying to prove that $x^2 + x + 1 = y^2$ has no integer solution, but I'm having a lot of trouble. So far I've tried proof by contradiction, but all of that seems to rely on me being able to ...
3
votes
2answers
110 views

The Conjecture That There Is Always a Prime Between $n$ and $n+C\log^2n$

Let $n$ be 113. Use $n+C\log^2n$ to find the next consecutive prime or at least approximately how far away it is. Will you show me how to work this out step by step to show me how to use this formula? ...
0
votes
2answers
20 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
1
vote
1answer
33 views

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$.

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$. My Attempt: $(\rightarrow):$ $$\alpha i=ai-b-ck+dk \Rightarrow -b+ai+dj-ck$$ ...
0
votes
1answer
24 views

Differentiation proof

Find the co-ordinates of the point on a curve $y=x^2+3x-1$ at which it is parallel to the line $ y=5x-1?$ unsure how to solve this
0
votes
1answer
44 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
2
votes
2answers
26 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
4
votes
2answers
104 views

Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$.

Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$. How do I prove this statement? I'm new to proofs, and I find this to be too obvious to prove. Please help me
0
votes
2answers
45 views

If $c | ab$, then $c | a$ or$ c | b$

I need help proving/disproving the implication, If $c | ab$, then $c | a$ or $c | b$ So far, I got Assume $c | ab$ then $ab= cl$ for some integer $l$ Now what should my next step be?
0
votes
2answers
54 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
0
votes
3answers
55 views

prove $m^{m-1} < (m-1)^m$ for m > 3

I found that if m > 3 then $m^{m-1} < (m-1)^m$ for m > 3 seems to hold true for a lot of cases. Can someone prove this inductively ?
1
vote
1answer
34 views

Prove that $ f\left(\bigcup _{i\in I}A_i\right)=\bigcup_{i\in I}f(A_i)$

Let $f \colon A\rightarrow B$ be a function. Furthermore let $I$ be a set and $\forall i\in I,A_i\subseteq A$ (1) $\displaystyle f\left(\bigcup_{i\in I}A_i\right)=\bigcup_{i\in I}f(A_i)$ (2) ...
0
votes
2answers
80 views

Any power of logarithm is $O(N)$

This is more of a computer science question but it uses calculus and proof techniques so I think it might be more appropriate here. Basically, how do I prove that, for any constant $K \geq 1$, ...
8
votes
6answers
812 views

Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational?

Let $x$ be rational with $0<x<1$ and let $y$ be the rational defined by $y = 1 - x.$ Let $n$ be any natural number with $n>2.$ Then I want to prove that $$x^{(1-1/n)}+ y^{(1-1/n)}$$ will ...
1
vote
1answer
51 views

How to write a general proof to prove that for all $m$, $m^n \geq n^m$

After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$. In other words, this can be generalized as: For ...
0
votes
0answers
33 views

Proving Catalan recurrence with Strong Induction

I am a student currently learning Strong Induction in my theory of computation course. I am stuck on a question and don't have a clue on how to start. I know how the principle of strong induction ...
1
vote
1answer
39 views

Proving a Special Case of a Limit Theorem

I'm having trouble proving a special case of the limit theorem below. I attempted a proof by contradiction that appears to me to make sense in the first direction but I'm not able to come up with ...
0
votes
1answer
36 views

Big Omega Proof for $5^n = \Omega(6^n)$

I got the Big-O ($\mathcal{O}$) proof but I am having troubles with the Big Omega ($\Omega$) proof. I am trying to prove $6^n$ is not the tightest bound for $5^n$. Is this logic true: $7^n$, $8^n$ ...
2
votes
3answers
87 views

Bijective Proofs

Give a bijetive proof: The number of subsets [n] equals the number of n-digit binary numbers. I do not understand how to do this problem, can someone help me figure it out? I have this so far, Let ...
18
votes
1answer
663 views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
1
vote
1answer
37 views

How to “prove” the following very simple vector identities

For my upcoming exam, I have to be able to prove the following four statements (among a bank of others). Though they are pretty obviously true, I'm struggling to come up with a way to prove them ...
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votes
3answers
46 views

Prove $ f(A1) \setminus f(A2) \subseteq f(A1 \setminus A2) $

here is my proof but it only works for functions that have an inverse:
0
votes
1answer
26 views

Extending a theorem true over the integers to reals and complex numbers

How does one generally extend a theorem proved over the integers to the real numbers and beyond e.g. induction proofs, De Moivre's Theorem? I am aware that to extend a theorem proved over ...
0
votes
3answers
52 views

Show that 1 is the supremum of $S = \{ x \epsilon R: x^2 < x \}$

I'm still new to proof writing so I was wondering if I could have a little help organizing my thoughts on this, I attempted this proof in a slightly oblong way that is probably not the standard, but I ...
0
votes
1answer
32 views

rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until…

"rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until you have an equation with $\sqrt{5}$ on the left and a ratio of two expressions involving $\sqrt{5}$ on the right." Ok..All i need to know is if i'm ...
2
votes
1answer
31 views

Finding a bound on a specific value of a holomorphic function

Let $f$ be a holomorphic function on $\overline{D(0,1)}$ such that $|f(z)| \leq M,$ if $|z|=1$ and $Im(z) \geq 0$ and $|f(z)| \leq N, $ if $|z|=1$ and $Im(z) <0.$ Could anyone advise me how to ...
-4
votes
2answers
32 views

Suppose S and T are two sets. Prove that if (S ∩ T) = S then S ⊆ T

I'm new to this entire proof thing, and I am so confused Suppose S and T are two sets. Prove that if (S ∩ T) = S then S ⊆ T Please help me
0
votes
1answer
25 views

If $a \in \mathbb{N}$, prove that gcd$(a, a+2)$ is $1$ if $a$ is odd and $2$ if $a$ is even.

Once again the problem is: If 'a' is an element of N, prove that gcd(a, a+2) is 1 if 'a' is an odd number, and 2 is 'a' is an even number. I really have no idea on how to prove this, and I'm brand ...
0
votes
1answer
23 views

Introduction chapter Exercise Q3 from “How to Prove It: A Structured Approach”

The following question is from the book "How to Prove It: A Structured Approach" Second Edition. Theorem 3 : There are infinitely many prime numbers. Euclid's proof Introduction Chapter : Exercise ...
0
votes
3answers
51 views

Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
6
votes
4answers
1k views

There exist no integers for which $x^2-4y=2$

I am working on a new exercise in my textbook: $$\text{Prove that: (P): }\;\nexists \;x,y \in \mathbb{Z}, x^2-4\cdot y = 2 $$ I am stuck and I would really like to see a correct proof so I can move ...
1
vote
1answer
111 views

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets I found this proof on a certain web page A direct proof would be ...
0
votes
1answer
33 views

Show that are logically equivalent

Can you answer me , please. 1- Show that (p→r)∧(q→r) and (p∨q)→r are logically equivalent 2- show that (p → r) ∨ (q → r) and (p ∧ q) → r are logically equivalent without truth table .
2
votes
2answers
32 views

Induction Proof without Explictly Using The Induction Hypothesis?

I have encountered several problems where one can prove the desired result without actually needing the induction hypothesis. More specifically, you basically just pick $n \in \mathbb{N}$ and run ...