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
3answers
67 views

Give an infinite collection of intervals such that $A_{n+1} \subset A_n$ and $\cap^{\infty}_{n=1} A_n = \varnothing$

Give an example of an infinite collection $A_n $ of intervals such that $A_{n+1} \subset A_n$ and $\bigcap^{\infty}_{n=1} A_n = \varnothing$. I have come to the following collection of intervals, ...
0
votes
1answer
21 views

prove that a sequence that tends to a limit, multiplied by n, the sequence will be kl.

Prove that if a sequence $s_n$ goes to a limit L as $n \rightarrow \infty $, then for a number $k > 0 $ then the sequence ${kn}$ will tend to the limi $kl$. Is this simply because k is isolated ...
0
votes
1answer
41 views

Showing that a set $D$ is closed and open

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) \mid |t-t_0|\leq T, |u-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
0
votes
1answer
2k views

Mathematical Induction Factorials, sum r(r!) =(n+1)! -1 [duplicate]

How do I prove that $$\sum\limits_{r=1}^{n} r(r!) = (n+1)!-1$$ I was able to get to factor: $LHS = k(k!) + (k+1)(k+1)!$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, RHS = (k+2)! -1$
4
votes
1answer
505 views

Prove that the only prime triple is 3, 5, 7 [duplicate]

Prove that the only prime triple is $3,5,7$. I tried proving using this method: Multiplication of $3$ jumps back and forth between being an even and an odd number. Thus goes from odd to odd over an ...
2
votes
1answer
56 views

Help with understanding a proof on Ordinary Differential Equations

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) | |t-t_0|\leq T, |u(t)-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
0
votes
1answer
117 views

Prove that any odd number can be expressed as $4n+1$ or $4n+3$

Prove that any odd number can be expressed as $$4n+1$$ or $$4n+3$$ I can see that this is true, but I am not certain on how to make a formal proof.
1
vote
2answers
79 views

Let $a,b,c \in \mathbb{R^+}$, does this inequality holds $\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$?

Does the following statement/inequality holds for $a,b,c \in \mathbb{R^+}$? $$\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$$ I've been thinking for hours and I ...
0
votes
2answers
73 views

Prove that $3n +5m = 12$ for any two natural numbers.

Prove that $\exists n,m \in \mathbb{N}$ such that $$3n+5m=12 $$ This is clearly false, but I am not sure how to conduct a proof stating it is false. Should I just give examples with $n = 1,2,3$ and ...
2
votes
0answers
87 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
710 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
324 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
63 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
90 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
75 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
174 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
512 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
64 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
68 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
1answer
101 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
739 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
226 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
64 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
65 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
53 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
149 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
86 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
69 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
98 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
132 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
84 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
161 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 ...
0
votes
0answers
25 views

Discrete Math Equation Proof (by induction?) [duplicate]

Consider the following description of a game. There are n people playing, one of whom leads the game. They are playing on a playing field with no obstacles. Everyone carries one water balloon. ...
-2
votes
3answers
62 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
63 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
130 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
40 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
77 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
443 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
153 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
224 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
108 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
103 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
102 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
61 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 ...