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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2answers
69 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
37 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
20 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
32 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
17 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

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
63 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
112 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
12 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
44 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
15 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
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2answers
44 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
31 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
37 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
29 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
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2answers
72 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 ...
1
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3answers
96 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
33 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
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2answers
26 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
41 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
63 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 ...
7
votes
2answers
54 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
1answer
77 views

How learn proofs? [closed]

I'm in high school, and I'd like to know how you guys learn proofs? What method and attitude you guys take when learning proofs? For example, when learning the proof of something simple like the sum ...
1
vote
2answers
22 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 ...
1
vote
0answers
14 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
13 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
45 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
30 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 ...
0
votes
0answers
29 views

How to prove a congruent to b (mod n) is a bijection?

I can prove it's an equivalence relation, but NO idea how to prove it's a bijection. I know I need to prove it's surjective/injective, but how do I establish it to even be a function?
1
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3answers
46 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
19 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
74 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
23 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
29 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
155 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
27 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)]
0
votes
1answer
25 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.
1
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3answers
38 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
43 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
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3answers
52 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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3answers
86 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
64 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
22 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
91 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
43 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
69 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
54 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
46 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
47 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
26 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
45 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 ...