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.

learn more… | top users | synonyms

0
votes
2answers
76 views

How to prove that $\lim_{x \to \infty} x = \infty$

Please refrain from using logic symbols, as I do not understand those. So, this is the question: $$\lim_{x \to \infty} x = \infty$$ Proving this using the actual formal definition of a limit. So ...
2
votes
3answers
47 views

Proving if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$

This is one of the problem I have been solving in Velleman's How to prove book: Prove that if $A \subseteq B$ and $A \nsubseteq C$ then $B \nsubseteq C$ This is my solution: Suppose $A ...
2
votes
1answer
24 views

Point in a rectangle

$ABCD$ is a rectangle and $P$ is a point in the same plane. If the perpendicular through $C$ to $AP$ and the perpendicular through $B$ to $DP$ intersect at $Q$, prove that $PQ \parallel AD$. ...
3
votes
1answer
38 views

Show g is unbounded above if g and g' are increasing

Suppose $g$ is a function defined on the set of real numbers where $g(y)$, $g'(y)$, and $g''(y)$ are all greater than $0$ for all $y \in \mathbb R$. Show that $g$ is unbounded above as $y$ approaches ...
2
votes
2answers
33 views

Proving that median of list $[x_1,x_2,…,x_n]$ minimises the sum $\sum_{i=1}^{i=n} |x_i-m|$ where $m$ is some number [duplicate]

The problem is in the title. Here is a detailed description: Let's say we have list $[x_i]_{i=1}^{i=n}$ where $x_i\in\Bbb{N}$. I want to pick such $m\in\Bbb{N}$ which minimises the sum ...
1
vote
4answers
68 views

Integrals are equal

Suppose that $f$ is integrable on $[a, b]$. Prove that there is a number $x \in [a, b]$ such that $$\int_a^x f(t)\,dt = \int_x^b f(t)\,dt .$$ Show by example that it is not always possible to choose ...
3
votes
2answers
125 views

Rolle's Theorem with roots

Let $f : [a, b] \to \mathbb R$ be $n$ times differentiable and have $n+1$ distinct roots (i.e. solutions of $f(x) = 0$) in $[a,b]$. Show that there is an $x \in [a, b]$ such that the $n^{\text{th}}$ ...
0
votes
1answer
14 views

Infinite sequence of real numbers converging to x and y

So the question is: Suppose $x_i$ and $y_i$ are infinite sequences of real numbers converging to x and y. Show that $(x_i + y_i)$ converges to $x+y$. Show that $x_iy_i$ converges to $xy$. Here's ...
3
votes
2answers
49 views

Inverse using Fundamental Theorem of Calc

Find $(f^{-1})'(0)$ if $f(x) = \int_1^x{ \cos(\cos t)dt}$ So question about this. For the problem there was no interval given so that the function $\cos(\cos t)$ was strictly increasing (which we ...
0
votes
1answer
21 views

How to find the characteristic polynomial of this transformation?

Let V be a finite-dimensional inner product space, and let W ⊂ V be a subspace. Let T : V → V be the linear transformation “orthogonal projection onto W”: T(x) = ProjW x. Show that T is ...
2
votes
2answers
55 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: ...
1
vote
1answer
37 views

Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
0
votes
2answers
47 views

Sum of odd numbers is odd if each of the natural numbers is odd

The question is: Proof that the sum of an odd number of natural numbers is odd if each of the natural numbers is odd. Here's what i tried already but it didn't work: $\sum_{i=0}^n i = 2n-1$ but ...
0
votes
1answer
32 views

interval proof using points

So this is for my advanced calculus class (Real Analysis II) which is a proof class. The question is: If $a<b$ are points in an interval $D$, show that $[a,b]$ $\subset$ $D$. I feel like its ...
0
votes
1answer
86 views

Use the Mean Value Theorem to show that if $|f'(x)| ≤ C<1$, then $f$ has at most one fixed point

Use the Mean Value Theorem to show that: if $|f'(x)| ≤ C < 1$ $\forall x$, then $f(x) = x$ has at most one solution. So using the Mean Value Theorem I know that $$-1<-C\leq ...
2
votes
3answers
109 views

Continuous functions and infinum

Let $f:\mathbb R \to \mathbb R$ with $f(-2)=4$ and $f(3)=7$. Let $S:=\{x \in [-2,3]\mid f(x)\geq 5\}$. Then $\alpha:=\inf S$ exists. If $f$ is continuous at $\alpha$, show that: (a) ...
0
votes
2answers
35 views

I didn't figure out how the result in part (i) can help in (ii). Anyone has any idea??

The determinant turns out to be -3 in part (i) How can this help in showing that the 4 vectors in the end are linearly independent?
0
votes
2answers
44 views

The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...
4
votes
2answers
104 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
3
votes
3answers
76 views

Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism

I'm working on the following problem: Let $G$ be a group. Prove that $G$ is abelian if and only if $\varphi(g) = g^2$ is a homomorphism. My solution: First assume that $G$ is an abelian group ...
8
votes
2answers
69 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
vote
2answers
25 views

Prove proposition on real numbers and uniqueness.

How would I go about proving the following proposition. Do I have to prove uniqueness, or that if $x^2 = r$, then $x = \sqrt r$? Prove given any $r \in \mathbb R\gt 0$, the number $\sqrt r$ is ...
2
votes
1answer
40 views

Proof of Polyates Lemma

In Sbiis Saibian's site I came across Polyates Lemma which states that $$(b \uparrow^k m) \uparrow^k n\ <\ b\uparrow^k (m+n)$$ for all positive integers b,m,n,k with $b\ge 2$ and $k\ge 2$. He ...
0
votes
1answer
17 views

Approximating a field by perfect fields.

Let's consider an arbitrary field $K$ and raise the following question: in which sense can we approximate $K$ by a perfect field? Any reasonable notion of approximation by a perfect field should admit ...
0
votes
3answers
55 views

Help explain the end of this proof for infinitely many primes?

by contradiction, assume finitely many primes $p_1, p_2,\cdots, p_k$. let $N = p_1p_2\cdots p_k + 1$. Note $N > 1$. Now, by the fundamental theorem of arithmetic, there exists a number $p_j$, where ...
0
votes
1answer
36 views

How would you solve these similar logic problems?

I'm not sure how to derive the conclusion from this problem (x)(Ax ⊃ Bx) Am & An / Bm & Bn As well as a similar problem with a disjunction instead of a conjunction (x)(Ax ⊃ Bx) Am v ...
1
vote
3answers
49 views

Prove if $x ∈ \mathbb{R}$, such that $0 ≤ x ≤ 1$, and $m,n ∈\mathbb{ N}$, with $m ≥ n$. Then $x^m ≤ x^n$

How to prove the following prop. Let $x \in \mathbb{R}$, such that $0 \le x \le 1$, and $m,n \in\mathbb{ N}$, with $m \ge n$. Then $x^m \le x^n$. I don't exactly know where to begin with this proof, ...
-1
votes
1answer
25 views

Prove proposition on real numbers

How would I go about proving the following proposition. Thanks. If $r < 0$ there exists no $x \in\Bbb R$ such that $x^2 = r$.
0
votes
2answers
79 views

Show that a function is constant

Let $S$ be a non-empty set of real numbers such that if $a,b$ are distinct elements in $S$, then $|a-b|\geq 1/2014$. Let $f:\mathbb R \to \mathbb R$ be such that the range of $f$ is a subset of $S$. ...
0
votes
2answers
27 views

Proof on maps and basic set theory

I am not sure about this question so I figured I would ask it on here. The question is: List all maps $\psi$ from $S = \{1,2\}$ to $T =\{-1,-2\}$ such that $\operatorname{Im}\psi = T$. Is the ...
1
vote
1answer
37 views

How would one derive conclusions from this?

Using only the 18 rules of inference without CP or IP derive the conclusion (x)(Ax ⊃ Bx) (x)(Bx ⊃ Cx) / (x)(Ax ⊃ Cx) As well as when using an Existential Quantifier (x)(Bx ⊃ Cx) (∃x)(Ax & ...
12
votes
3answers
191 views

Prove that $\frac{(p^{n}-1)(p^{n}-p)…(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Apparently this question requires a method linked with linear algebra but I was wondering if it was possible to solve it in a formal way like an induction on $n$ or by using an identity for $p^{n}-1$ ...
1
vote
1answer
30 views

Using CP prove the truth of a tautology

Having trouble figuring out this tautology using CP and the rules of infrence [P ⊃ (Q ⊃ R)] ≡ [Q ⊃ (P ⊃ R)]
1
vote
1answer
44 views

Semantic tableau software

Is it possible to find software to perform semantic tableaus (as described in http://en.wikipedia.org/wiki/Method_of_analytic_tableaux) automatically? Right now I am proofing it by hand.
2
votes
3answers
46 views

Compare inequalities in a proof by induction

I am solving a proof by induction example. But I ended up with my hypothesis $$ a_{n-1} \geq \frac{2^n}{2}+n^2-2n+1 $$ and my inductive step $$ a_{n-1} \geq \frac{2^n}{2}+\frac{n^2}{2}-\frac{n}{2}. ...
1
vote
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!
0
votes
3answers
373 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
votes
3answers
91 views

proof about limits of functions

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x) and f''(x)$ are all positive for each $x \in \mathbb R$. Show that $\lim_{x \to \infty} f(x)=\infty$. So $f''(x)$ is the second derivative of ...
5
votes
1answer
77 views

Convergence of sequence of $L^{p}$ function

Given that $\Omega \subset \mathbb{R}^{n}$ is bounded. If you are given that $u_{k} \rightarrow u$ in $L^{p- \epsilon}(\Omega)$ and a functions $f: \mathbb{R} \rightarrow \mathbb{R}$ where ...
0
votes
1answer
37 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 ...
3
votes
3answers
130 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
vote
1answer
52 views

Vector analysis - Curl of vector

How to prove it? I have tried several times to solve it, but I still get stuck everytime.
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
votes
5answers
259 views

Basic set theory proof about cardinality of cartesian product of two finite sets

I'm really lost on how to do this proof: If $S$ and $T$ are finite sets, show that $|S\times T| = |S|\times |T|$. (where $|S|$ denotes the number of elements in the set) I understand why it is true, ...
2
votes
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
vote
1answer
170 views

proving gradient of a function is always perpendicular to the contour lines

Can someone give an explanation of how such a proof would go, given a function example: $y = f(x)$
0
votes
1answer
30 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
2
votes
1answer
51 views

Maximum load when placing N balls in N bins

In an academic paper I am reading the following.. When $n$ balls are placed into $n$ bins (each ball being placed into a bin chosen independently and uniformly at random) with high probability, the ...
0
votes
2answers
52 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
votes
0answers
28 views

$P+Q:=\varphi_{O}^{-1}\left(\varphi_{O}(P)+\varphi_{O}(Q)\right)$

let X is affine space and $\overrightarrow{X}$ is vector space associted to X $$\begin{array}{ccccc} & \varphi_{O} : & X & \longrightarrow & \overrightarrow{X}\\ & & ...