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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2
votes
0answers
93 views

Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
0
votes
2answers
999 views

Proving a Sequence Does Not Converge

I have a sequence as such: $$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$ Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact? ...
2
votes
1answer
45 views

The limit of a sequence when at $n-1$

Suppose $\sum\limits_{n=1}^{\infty} a_n$ is a series that converges. Therefore, $\lim\limits_{n \to \infty} S_n$ exists, where $S_n$ is the sum of the first $n$ terms of the series. So, let ...
0
votes
2answers
368 views

Proof that changing a finite number of terms in a series does not change where or not it converges

I want to prove the following theorem: Changing a finite number of terms in a series does not change whether or not it converges, although it may change the value of its sum if it does converge ...
1
vote
1answer
71 views

Does the principle of complete induction imply the well-ordering principle

Proof: Assume the PCI. Let $T$ be a nonempty subset of $\mathbb{N}$. Then $T$ has some element $x$. Then $\{1,2,...,x-1\}$ is a subset of $\mathbb{N} - T$. By the PCI, $x$ is an element of ...
0
votes
3answers
95 views

Question on Proof of the Contraction Mapping Theorem

Contraction Mapping Theorem If $T\colon X\to X$ is a contraction mapping on a complete metric space $(X,d)$ then there is exactly one solution $x\in X$. Proof: Let $x_0$ be any point in $X$. We ...
0
votes
2answers
94 views

Proof of if $A \times B = A \times C$, and $A \neq \varnothing$, then $B=C$

Proof: suppose $A \times B = A \times C$ Then $\frac{A \times B}{A} = \frac{A \times C}{A}$ Therefore $B=C$ Is this proof valid?
2
votes
1answer
76 views

Proof of $gcd(f_{n},f_{n+2})=1$ for natural numbers

I'm going to use the Principle of Mathematical Induction to prove the above statement. Base cases: $(n=1)$ $f_{1}=1, f_{3}=2$ so $gcd(1,2)=1$ $(n=2)$ $f_{2}=1, f_{4}=3$ so $gcd(1,3)=1$ Assume that ...
0
votes
3answers
175 views

If $n$ is an odd integer, then there exist integers $a$ and $b$ such that $n=a^2-b^2$. [duplicate]

If $n$ is an odd integer, then there exist integers $a$ and $b$ such that $n=a^2-b^2$. Am I supposed to use induction or a direct proof?
0
votes
1answer
575 views

Prove that f is a constant function

Let $f$ be a function defined on R and suppose that there exists $M>0$ such that for any $x,y∈R$, $|f(x)-f(y)|≤M|x-y|^2$. Prove that $f$ is a constant function. I don't even know how to start, I ...
0
votes
1answer
66 views

if neither f nor g is differentiable at x=a. is $f+g$ differentiable at $x=a$?

Suppose that $f$ and $g$ are defined on R and that neither f nor g is differentiable at x=a. prove or disprove: f+g is not differentiable at x=a. I know how to show if f and g are differentiable at ...
1
vote
0answers
74 views

Induction in the caculus of terms - Mathematical Logic

I'm studying logic from the Ebbinghaus's book "Mathematical Logic" and when I tried to solve some of the exercises doubt rises. Given a calculus C consisting of the following rules: ...
1
vote
2answers
131 views

Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ is a continuous function using epsilon-delta.

THE QUESTION: Use the metric $(x,y)$ = $\rho(x,y)=|x-y|$ for the reals and use the metric $\rho((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ for the plane. Define $f:R\times R \to R$ as ...
1
vote
0answers
936 views

Proof of floor and ceiling functions

By definition: $ \lfloor {x}\rfloor = i \Rightarrow i \le x \lt i + 1 $ (floor function) and $ \lceil {x} \rceil = j \Rightarrow j - 1 \lt x \le j $ (ceiling function) So, how is the proof that ...
5
votes
4answers
244 views

Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$

Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$. So, I can intuitively see that this is true. My proof mostly makes ...
0
votes
4answers
65 views

Proof: For all real numbers $x$, If $x^2 - 5x + 4 \geq 0$, then either $x ≤ 1$ or $x ≥ 4$.

I need some help in proving the following statement: $x$, If $x^2 - 5x + 4 \geq 0$, then either $x ≤ 1$ or $x ≥ 4$. It would be greatly appreciated if someone could provide me a generic proof! I'm ...
3
votes
3answers
67 views

Strong induction doesn't require a base case?

I'm considering the natural numbers to be the nonnegative integers. The principle of strong induction can be stated as follows, "If $P$ is a property such that for any $x$, if $P$ holds for all ...
0
votes
0answers
56 views

Formalization of the Proof of the Theorem of the Bijection of Composition of Two Mappings

I'm trying to formalize in FOL the proof of the stated theorem. Assume two mappings $f$,$g$. With a slight circularity for brevity's sake, let $B(f): \text{"f is a mapping which is bijective whose ...
0
votes
0answers
54 views

Is this GCD proof valid?

I came across this theorem and wrote a proof, but I'm not sure if I made any incorrect assumptions. I also know that this isn't the easiest way to prove it - I just want to know if it works and ...
0
votes
2answers
45 views

Prove that for every $x \in \mathbb{Z}$ for which $x \geq 3$, if $x \equiv 3 \pmod 4$ then $3^x - 2 \equiv 0 \pmod 5$

Prove that for every $x \in \mathbb{Z} \geq 3$, if $x \equiv 3 \pmod 4$ then $3^x -2 \equiv 0 \pmod 5$. I was trying to use induction: Base case $(x = 3)$: If $3 \equiv 3 \pmod 4$ then $3^3 - 2 ...
0
votes
2answers
153 views

What approach should I take to establish this logical proof?

I need to design a logical math proof: Write a detailed structured proof to prove that if m and n are integers, then either 4 divides mn or else 4 does not divide n. Hint: Think about the form of ...
0
votes
3answers
90 views

Proof by cases, inequality

I have the following exercise: For all real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \leq 1$ or $x \geq 4$. I need you to help me to identify the cases and explain to me how to ...
1
vote
2answers
2k views

Prove that if an integral is 0, the function is 0 across that interval (for f(x) >= 0)

Assume f:[a,b] $\Rightarrow \mathbb{R}$ is continuous and f(x) >= 0 for all x $\in$[a,b]. Prove that if $\int_a^b f dx$ = 0, then f(x) = 0 for all x $\in$ [a,b]. My attempt at a proof a little ...
3
votes
1answer
73 views

The union of a countable set of countable sets is countable

Here is the proof provided in my lecture notes: Let $A = \{B_n | n < \omega =\mathbb{N}\}.$ Assume each $B_n$ is countable. For each $n < \omega,$ let $E_n$ be set of all bijections between ...
1
vote
5answers
107 views

Prove that for every $n∈N$ the expression is divisible by $10$?

Prove by induction: $3^{(4n+2)} + 1$ is divisible by $10$. My basic step: $3^{(4n+2)} + 1$, where $n = 1$ gives me $3^6 + 1 = 730$, which is divisible by $10$. However, then I have to do the ...
2
votes
1answer
138 views

Proving a Property of a Set of Positive Integers

I have a question as such: A set $\{a_1, \ldots , a_n \}$ of positive integers is nice iff there are no non-trivial (i.e. those in which at least one component is different from $0$) solutions ...
0
votes
1answer
86 views

Finding a combinatorial proof of this identity: $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$

Can someone prove this. Let $D_n$ be the number of derangements of $n$ objects. Find a combinatorial proof of the following identity: $$n!=\sum_{i=0}^n \binom{n}{n-i}D_i$$
2
votes
1answer
165 views

Prove that $\lim_{\Delta x\to 0} \frac{\Delta ^{n}f(x)}{\Delta x^{n}} = f^{(n)}(x).$

If $\Delta f(x)=f(x+\Delta x)-f(x)$, $(a)$ prove that $$\Delta\{\Delta f(x)\}=\Delta^2f(x)=f(x+2\Delta x)-2f(x+\Delta x)+f(x);$$ $(b)$ derive an expression for $\Delta^n f(x)$ where $n$ is any ...
-2
votes
3answers
64 views

Theorem proof of this equation

How would you prove the theorem $(-a)\cdot (-x)=ax$? If you used multiplication and addition axioms.
0
votes
1answer
64 views

professionally writing proofs

I am writing a proof for the Theorem (x-a)(x+a)=x^2-a^2 and directly proved it by manipulating the equation using multiplication and addition axioms. But I'm not sure what should be included in the ...
0
votes
1answer
147 views

Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
0
votes
1answer
41 views

How can I prove that this function is uniformly continuous?

How to show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous? Thank you. Also, how do I become good at writing these proofs?
1
vote
1answer
80 views

Prove Heine-Borel Thm

Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover." Suggestions: Let $S \subset \mathbb{R}$. If every open cover for ...
2
votes
1answer
495 views

Proving if the limit of $f(x)$ approaches zero, then the limit of $1/|f(x)|$ approaches infinity.

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...
2
votes
1answer
159 views

Open Cover for a Compact Subset

I am doing some extra exercises for an Analysis class, and I found this one. We haven't seen much of what an open cover is, but I want to learn it. So, here it goes, and thank you everyone! Let ...
2
votes
2answers
244 views

If a function $f$ is continuous in $[a,∞)$ and finite $\lim_{x→+∞}⁡f(x)$ exists, then it's uniformly continuous in $[a,+∞)$. [duplicate]

Prove that if $f$ is defined and continuous in $[a,+∞)$ and if there exists a finite limit $\lim_{x→+∞}⁡f(x)$, then $f$ is uniformly continuous in $[a,+∞)$ I know that since there exists a finite ...
2
votes
2answers
112 views

Proof of A is orthogonal $\Leftrightarrow \|Ax\|=\|x\|$

I want to proof this property. A is orthogonal $\Leftrightarrow \|Ax\| = \|x\|$ I tried to elaborate from this, but cannot see how to get any further: $\|A\| \times \sqrt{<x,x>}$ I ...
1
vote
1answer
111 views

Show that if x divides a power of 2, then x is a power of 2

I'm trying to prove that if $x$ divides $2^a$ for some integer $a \geq 0$, then $x = 2^b$, where $a \geq b$. In other words, if $x$ divides a power of 2, then $x$ is a power of 2. This makes sense, ...
0
votes
1answer
135 views

Strong induction definition clarification

I have a general question about strong induction: Assuming that the base case is 0, if I let my inductive hypothesis be that for all 0 <= k < n some statement is true, and if I prove that that ...
0
votes
1answer
64 views

Rooted Trees & Induction

So I am a little stumbled upon this question: A full binary tree is a rooted tree where each leaf is at the same distance from the root and each internal node has exactly two children. Inductively, a ...
2
votes
1answer
207 views

Question: Prove that a set of connectives is incomplete using structural induction

The proof generally begins with an inductive definition of the set. For example, let's say the set of connectives was {$\oplus$}. Let F be the smallest set such that: 1) Any propositional variable is ...
1
vote
6answers
111 views

Proof of $(\forall x)(x^2+4x+5 \geqslant 0)$

$(\forall x)(x^2+4x+5\geqslant 0)$ universe is $\Re$ I went about it this way $x^2+4x \geqslant -5$ $x(x+4) \geqslant -5$ And then I deduce that if $x$ is positive, then $x(x+4)$ is positive, so ...
1
vote
1answer
63 views

Noncontradiction behind the uniqueness proof and proof by mathematical induction

I'm walking through equivalences, as it appears, between $$\exists!x:P(x)\,\,{\overset{\mathrm{?}}{\equiv}}\,\,\exists x:P(x)\wedge(P(x_1)\wedge P(x_2)\rightarrow x_1=x_2),$$ where I am not sure what ...
0
votes
1answer
31 views

Prove that $Tu(x)$ is a contraction. $Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$

I want to show that $Tu(x)$ is a contraction where $$Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$$ and $$g(x,y) = \begin{cases} x(1-y) & 0\leq x\leq y\leq 1, \\ y(1-x) & 0\leq y \leq x \leq ...
0
votes
1answer
47 views

prove that |a| < b if and only if -b < a < b

I know I'd have to prove both sides here so: ...
0
votes
2answers
109 views

If a function is continuous and converges finitely, then it is bounded

Suppose that $f$ is a continuous function on $[a,+\infty)$ and that there exists a finite limit $\lim \limits_{x \to +\infty}⁡ f(x)$, then $f$ is bounded. I know that from the assumption I can ...
1
vote
1answer
316 views

Uniform continuous function proof

Suppose that $f$ is defined and continuous on $[a,b]$. Prove that $f$ is uniformly continuous on $(a,b)$ if and only if it is uniformly continuous on $[a,b]$ I know this is true because the open ...
0
votes
2answers
52 views

Interesting question posted earlier by another user need help solving

I've been trying to solve a problem a user posted that I thought was interesting. Considered a lucky number, the Thai government decides to issue coins of 9 baht. Show that, forall suciently large ...
-1
votes
3answers
132 views

Prove $2^n > 10n^2$ for all sufficiently large integers n.

How do I prove $2^n > 10n^2$ inductively? I know you can prove this to be true using calculus (i.e. taking derivatives). But how would I do it inductively?
4
votes
5answers
217 views

$A \oplus B = A \oplus C$ imply $B = C$?

I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both. But when it comes to solving ...