For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1answer
51 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
1
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1answer
46 views

Could you expand a little on this proof or Floyd-Warshall Algorithm?

I'm reading this. $\quad$ He gives a proof of Floyd-Warshall's algorithm but I don't understand what he's doing nor why it proves that. I can see an intuitive proof in my mind that is as ...
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2answers
85 views

How to find the limit of the sequence given by $X_{n+1} = \frac12 X_n + \frac1{X_n}$

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1{X_n}$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still ...
1
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1answer
47 views

Prove center of a group is a subgroup using one-step subgroup test

I'm not sure if this is correct. It doesn't seem so. If $a,b \in C$, then we must show $ab^{-1} \in C$. $$ab^{-1}x=axb^{-1}=xab^{-1}$$ This doesn't seem correct. I've seen two-step subgroup tests ...
2
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2answers
83 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
2
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3answers
58 views

Why can't a direct proof be made backwards?

Say we have the following implication: $$\textit{Let $x\in \mathbb{Z}$. If $5x-7$ is even, then x is odd. }$$ The method used by my book to prove this implication is by means of a proof by ...
2
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1answer
86 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
3
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2answers
29 views

Given a set $A$, how do I prove that there exists a set of all sets $x$ such that $\bigcup x=A$?

I am working with Zermelo-Fraenkel axioms. Specifically, I am allowed to assume the Axiom of Pair, Axiom Schema of Comprehension, Axiom of Union, and Axiom of Power Set, etc. (not yet allowed to use ...
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2answers
50 views

How to prove $x^{\phi(m)+1}\equiv x\pmod{p}$ [duplicate]

How do I prove that $x^{\phi(m)+1}\equiv x\pmod{p}$ when $m=pq$, two distinct primes? I kind of have an idea that it involves Euler's Theorem but it doesn't seem to be working as well as I wanted it ...
0
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1answer
70 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
1
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1answer
47 views

show that a loopless graph $G$ contains a bipartite spanning subgraph $H$ such that $d_H(v) \ge \frac{1}{2} d_G(v)$ for all v $\in$ V.

The hint in the appendix of book says that bipartite subgraph with with largest possible number of edges has such a property, but I don't know how to use this hint! any help would be appreciated.
0
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1answer
39 views

Calculus proof attempt

Why does $$\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}$$ not yield the same result as $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\dot{x}}\right) \ ...
0
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1answer
31 views

second derivative of a parametric equation

can someone please explain how in the proof for the second differential of a parametric function we get from to ? how do we calculate $\frac {d}{dt}$?
0
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2answers
33 views

Induction proof for $x \le y $

So these are $x \in \{1 ... i\}$ and $y \in \{ i + 1 ... n\}$, for $i, n \in \mathbb N$. I want to prove it for every $x \le y $. I know it`s easy but the solution is escaping me. I have tried with ...
1
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1answer
56 views

Prove or disprove this lemma for Catalan Numbers

Prove or disprove that for all non-negative integers $n$ and $r$ with $r+1$ is less than or equal to $n$, $C(n,r+1)=C(n,r)\times\frac{n-r}{r+1}$.
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0answers
35 views

How to prove that all powers of two minus one have only 1's when in binary representation?

It just came to my mind that all powers of two, when represented in binary format, are composed of only 1's, not 0's. I can see some logic behind it, however I can't seem to come up with an actual ...
2
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2answers
58 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
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2answers
42 views

How does this proof that the square of an integer, not divisible by 5, leaves a remainder of 1 or 4 when divided by 5 work?

Below I have a part of a proof of the fact that the square of an integer, not divisible by $5$, leaves a remainder of $1$ or $4$ when divided by $5$. But I am wondering where does the part highlighted ...
8
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2answers
76 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
1
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1answer
29 views

Proving if $\frac{3x+1}{x-1}$ is onto?

So, I have this function: $f(x)=\frac{3x+1}{x-1}$. So, in proving if it is onto, then by definition, for every b in B, there exists an a in A such that $f(a)=b$. So, let's solve or a. We get: ...
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1answer
38 views

How many square root matrices?

I would like to see a proof of the following statement: A positive-semidefinite matrix has precisely one positive-semidefinite square root, which can be called its principal square root. I ...
1
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1answer
46 views

Consider the function $g(x)=xe^x$. Make and prove a conjecture about the $n^\text{th}$ derivative of $g$. [closed]

Please help on this homework problem I have in my Proofs class. My prof is really bad at explaining and I don't know how to answer this problem! Thank you!
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3answers
497 views

Prove that a rational number minus an irrational number must be irrational. [duplicate]

Please help with this homework problem I have! I don't know how to prove this.
0
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1answer
38 views

Can you use proof by contradiction inside simple induction?

During a proof using simple induction, I assumed P(k) is true. Now in order to show P(k+1) is true using P(k), can I do a proof by contradiction on P(k+1) and say P(k) would be wrong if P(k+1) is ...
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1answer
43 views

Can I prove these series with limit a using induction?

This is the equation: It is true for: E a normed space and $(a_n)_{n \in \mathbb N}$ a convergent sequence with limes a. $$s_k = \frac1k\sum^k_{n=1} a_n \rightarrow a$$ $a = \lim_{n\rightarrow ...
2
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1answer
63 views

Prove formula's tautology

Prove that a formula that only consists of variables, logical negation and logical equality, and in which all variables and negation appear for an even number of times, must be tautological.
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2answers
51 views

Formal negation of $((p\rightarrow q) \vee (q \leftrightarrow r)) \rightarrow q$

Can someone give me an outline for how I can negate the following? $((p\rightarrow q) \vee (q \leftrightarrow r)) \rightarrow q$
3
votes
2answers
89 views

Integer inequality: $x + y +z> a + b + c$ does not imply $xyz > abc$

Prove by contradiction that for any integers $x,y,z,a,b,c$ greater than $0$ such that $x+y>a+b$, it is not implied that $x\cdot y\cdot z>a\cdot b\cdot c$? Obviously this statement is true. ...
2
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2answers
50 views

Alternate way to Prove or disprove $6\mid n(n+1)(n+2)$

This is my proof, I'm wondering if I'm correct, and how to do without induction. My Work Basis Step $$\frac{(1)(2)(3)}{6} = 1$$ Inductive Hypothesis Assume that $\dfrac{k(k+1)(k+2)}{6} = d$ where ...
1
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3answers
519 views

Explain proof that any positive definite matrix is invertible

If an $n \times n$ symmetric A is positive definite, then all of its eigenvalues are positive, so $0$ is not an eigenvalue of $A$. Therefore, the system of equations $A\mathbf{x}=\mathbf{0}$ has no ...
1
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1answer
39 views

Let A⊆R be a set. Prove that A is bounded if and only if there is some M∈R such that M>0 and that |x|≤M for all x∈A.

I proved the above problem as follows but received feedback that I'm not certain I understand. Can someone help me determine where I went wrong in the proof? Let A⊆R be a set. Suppose M∈R where ...
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2answers
55 views

Proof for consecutive integers

Prove that if $n$ is an odd integer, $n^3$ is the sum of $n$ consecutive integers. I'm confused on how to prove something with consecutive integers.
0
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4answers
48 views

Induction Proof with a $\neq$ 1 [duplicate]

For $a \neq 1$ and $n>0$, $(1-a^{n+1})/(1-a)=1+a+a^2+...+a^n$ How do you prove this by induction?
2
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1answer
21 views

Prove an x exists with f(x) = f(x + T/2)

Suppose $f: \mathbb{R} → \mathbb{R}$ is a continuous and periodic function with period $T > 0$. Show that: there exists an $x \in \mathbb{R}$ such that $f(x) = f(x + T/2)$. We figured out we ...
1
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3answers
93 views

$f'(a)=0$ implies $x=a$ is not a simple zero of $f$

Let $a$ be the root of a polynomial $f(x)$ and let $f'(a)=0$. Then $x=a$ is not a simple zero of $f(x)$. What is the name of this theorem and does someone know a simple (high school level) proof?
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3answers
56 views

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime.

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime. From the context in which this question was set, I believe I have to prove/disprove it using ...
0
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1answer
43 views

Inductively showing $g(s) = 3(g(s-1)+g(s-2))+1$ is odd for all $s$

How would I show this equation is odd by using the induction hypothesis: $$ g(s) = 3(g(s-1))+(g(s-2))+1 $$ I was thinking that I would prove $g(s)$ is odd by $g(s+1) = 3(g(s)+g(s-1))+1$. How would I ...
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0answers
37 views

trouble in reading statements involving symbols

I have trouble in reading statements involving symbols. (1) For example, when reading a statement from a paragraph: Suppose the size of the set is n. The symbol ...
1
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1answer
25 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
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2answers
62 views

Prove that $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$

I'm trying to practice proof writing, and found the following text question: For all sets A,B,C: $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$ The first step I was thinking of showing is that: ...
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0answers
60 views

Assuming a Solution Exists

Students who are beginning to learn proofs (and some seasoned pros) occasionally commit the error of assuming what they're trying to prove. My question involves assuming at the onset that a solution ...
1
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1answer
34 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
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3answers
35 views

Disproving statement of convergence

How can I disprove this statement: If the sequence $(a_n+b_n)^\infty_{n=1}$ converges, then both $(a_n)^\infty_{n=1}$ and $(b_n)^\infty_{n=1}$ converge. Is this statement able to be disproved? I ...
2
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1answer
35 views

Trouble Understanding Proof About Polynomials

In the question I have to prove that: There is no polynomial $P (x) = a_n x^n + a_{n−1}x^{n−1} + · · · + a_0$ with integer coefficients and of degree at least 1 with the property that $P(0), ...
2
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3answers
265 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
0
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1answer
51 views

Prove the sequence $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge

How would I be able to prove that $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge? I know that for a sequence to converge to a limit, then for all $\varepsilon > 0, \exists N \in \mathrm N ...
0
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1answer
36 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
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1answer
64 views

Solve this tautology

Hypotheses: not $q$, $p$ or not $s$, $p \rightarrow$ ($d$ and $q$), $e \rightarrow s$ Conclusion: not $e$ I have thus far, but unsure how to proceed. I am looking forward to solve it using ...
0
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2answers
102 views

Prove that $x=0.1234567891011\cdots$ is irrational [duplicate]

Prove that $x=0.1234567891011\cdots$ is irrational Proof: we argue by contradiction.suppose x is rational. then its decimal expansion ultimatetly periodic. Lets p denote the perid of this expansion. ...
0
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1answer
43 views

Prove that a function is a linear transformation.

Lets say that I have a vector space $A$ and a linear transformation defined as $f : A → A$. Now I have a function $g : A → A$ defined as $g(a) = bf(a)$ where $a\in A$ and $b \in \mathbb{R}$ is a ...