For questions about the formulation of a proof, not about the mathematics behind it.

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2answers
91 views

Prove $3+ 5 \sqrt {2}$ is irrational

Prove $3+ 5 \sqrt{2}$ is irrational. I have some ideas about this proof, but I am not quite finished. I understand being irrational means the number would not be in the form of $\frac pq$. I have ...
2
votes
4answers
66 views

Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.

Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this: You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = \...
1
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1answer
30 views

proof the derivate of gamma function using the limit definition

using $\Gamma(z+1)=z\Gamma(z)$ and $\Gamma(z)=\lim\limits_{n\to+\infty}\frac{n!n^z}{z(z+1)\cdots(z+n)}$ proof that $$\psi(z+1)=-\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{1}{m}-\ln n\right)+\sum_{l=...
0
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1answer
54 views

The sum which gives $3^n$

So I have the following which I must prove : $$\sum_{(n_1,n_2,n_3)\,:\,n_1+n_2+n_3=n} \binom{n}{n_1, n_2, n_3} = 3^n$$ I'm not sure where I must begin. This is a multinomial.
0
votes
1answer
30 views

Prove a property of a function H based on the definition provided

Define $$H(n) = \begin{cases}{} 0 & n\leq 0\\ 1 & n = 1 \textrm{ or } n = 2\\ H(n-1) + H(n-2) - H(n-3) & n>2\\ \end{cases}$$ Prove $\forall n\geq 1$ that $H(2n) = H(2n-1) = n$. ...
0
votes
2answers
40 views

Proof (a | b and a not divide b) -> a not divide (b+c)

Prove $\forall a\in \mathbb Z, \forall b\in \mathbb Z, \forall c\in \mathbb Z, (a | b \land a\nmid c) \rightarrow a\nmid(b + c)$. Maybe a gentle nudge in the right direction
0
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2answers
44 views

Multiplying matrices / corresponding systems of equations

I'm having some trouble with a problem in linear algebra: Let $A$ be a matrix with dimensions $m \times n$ and $B$ also a matrix but with dimensions $n \times m$ which is not a null matrix. (That's ...
1
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1answer
33 views

Determining the exact one from all possible Jordan Canonical Forms of a matrix

Here is the example I encountered : A matrix $\ M\ $ $(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)^3.$ So considering the two possible sets of elementary ...
0
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1answer
22 views

Proof of an algebraic statement [duplicate]

Let $V$ be a $n$-dimensional vector space. Let's also say that we have two linear operators: $A,B\in L(V)$ and $AB=0$. Then how do I prove that the sum of the ranks of operators is smaller than $n$, i....
0
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2answers
46 views

Two matrix proofs

linear algebra problem I'm having some trouble wrapping my head around: Given two square matrices $A,B$ with dimensions $n\times n$ and that $A=I-AB$ : I've already proved with relative ease that $A$...
0
votes
0answers
40 views

If $\Sigma$ is the splitting field for $f$ over $K$ and $K\subseteq L \subseteq \Sigma$, show that $\Sigma$ is the splitting field for $f$ over $L$.

If $\Sigma$ is the splitting field for $f$ over $K$ and $K\subseteq L \subseteq \Sigma$, show that $\Sigma$ is the splitting field for $f$ over $L$. I believe the general idea of this proof is as ...
0
votes
2answers
46 views

How to prove a statement with two “ if and only if”

If $H$ and $K$ are subgroups of $G$, show that $HK$ is a subgroup if and only if $HK \subseteq KH$, if and only if $KH \subseteq HK$. This statement confuses me. Does mean I need to prove that $HK$ ...
3
votes
1answer
53 views

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$. So my plan was to prove this by contradiction. I'm wondering if there is a ...
0
votes
1answer
32 views

if $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$

If $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$ I proved that if $f$ is integrable on $[s,b]$ then $f$ is integrable on $[a,b]$ But how to prove the ...
3
votes
0answers
58 views

Is there any mistake in my proof?

My little brother started fiddling around with his calculator, and noticed something curious: $$ \Large \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\sqrt{a \cdot \ldots}}}} = a $$ So I went ahead and wrote a ...
-1
votes
5answers
146 views

Prove number of handshakes between $n$ people is $\tfrac{n(n−1)}{2}$ by induction [closed]

How do we calculate the number of handshakes between $n$ people? And where do I apply the inductive step?
4
votes
2answers
112 views

Number of Taxicab routes in a triangular city

I am assuming a triangle that is "almost" half a rectangular city with taxicab geometry. I am trying to find the number of paths in this triangular city. Assuming that the ride starts from the corner ...
1
vote
1answer
59 views

Suppose my progress is in Baby Rudin's chapter 4. Is it possible to discuss the uniform continuity of $x^t$ without using facts in later chapters?

Let $f: [0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=x^t$. Prove that If $t \in (0,1]$ then $f(x)=x^t$ is uniformly continuous on $[0, \infty)$. If $t \in (1, \infty)$ then $f(x)=x^...
1
vote
3answers
56 views

How to write a rigorous proof for this statement?

Prove that for finite set $X$, the function $f:X \to X$ is surjective if and only if it is injective I have the idea of proof in my mind but find it difficult to translate it into mathematical ...
1
vote
4answers
112 views

prove that $n(n+1)$ is even using induction

the base case of $n=1$ gives us $2$ which is even. assuming $n=k$ is true, $n=(k+1)$ gives us $ k^2 +2k +k +2$ while $k(k+1) + (k+1)$ gives us $k^2+2k+1$ whats is the next step to prove this by ...
0
votes
1answer
153 views

Proof that measure of variation proposed by Jordan is same as sample variance

This is the problem I had encountered Statistic $G_k$, defined for $k=1,2$ as $$G_k= \frac 1{n(n-1)}\sum^n_{i=1}\sum^n_{j=1}|X_i - X_j|^k$$ was proposed as a measure of variation by ...
2
votes
1answer
34 views

$\lfloor\log{p_{n}}\rfloor$ having more than one solution for individual $k$

Question: If you assume that a. $k\in\Bbb{N}$ b. $p_n$ denotes the $n$'th prime number. $p_0$ doesn't exist. c. $n\in\Bbb{N}$ I am fairly certain that: At least two distinct integer values for $...
2
votes
1answer
56 views

Any open cover of $S^1$ is an open cover of the annulas

The question goes like this : If $\{U_i:i\in I\}$ is an open cover of the unit circle in $\mathbb R^2$ then show that it is an open cover of an annulus $1-\delta\lt ||(x,y)||\...
1
vote
1answer
51 views

The countable dense subset of every compact metric space

Show that any compact metric space has a countable dense subset. I am having problem with finishing the proof after a few steps. This is how I am going : So, let $X$ be the compact ...
0
votes
1answer
33 views

Fixed “$\mathcal Set$” for continuous maps on compact spaces

Let $$f:X\rightarrow X$$ be a continuous map on the compact metric space $X.$ Show that there is a subset $A\subset X$ such that $f(A)=A$. Now the given hint is that to ...
0
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1answer
73 views

Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
-2
votes
2answers
37 views

Prove by induction that $\sum_{i=1}^n i \geq \frac{n^2}{2}$ [closed]

Can someone show me a formal proof of this exercise ? \begin{equation} \sum\limits_{i=1}^n i \geq \frac{n^2}{2}, \quad \forall n \in \mathbb{N}. \end{equation} Thanks to anyone who can help! :)
0
votes
1answer
20 views

Prove a relation of a distance function

I had to do an exercise with this function: $$ d_M:\Bbb C \rightarrow \Bbb R, \quad z \rightarrow inf\{ |z-w|; w \in M| $$ with $\emptyset \neq M\subset \Bbb C$. First I proved that this function is ...
0
votes
1answer
103 views

Proof that a certain language is Turing Decidable

$$L_1 = \{\langle R,S \rangle \mid \text{$R$ and $S$ are regular expressions and }L(R) \subseteq L(S)\}$$ $$L_2 = \{\langle M\rangle\mid \text{$M$ is a DFA that accepts $w^r$ whenever it accepts $w$} ...
1
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2answers
90 views

Epsilon-Delta proof for continuity

I have a lot of trouble figuring out how to work with this proof technique for continuity. The definition says: $$ \forall \varepsilon \space \exists \delta \quad |x-a|\lt \delta \quad \Rightarrow \...
1
vote
1answer
39 views

Looking to receive feedback on elementary proofs in topology

I'm looking to receive some feedback on a couple of proofs I wrote verifying the discrete and trivial topologies and another simple topology. I'm inexperienced with proof (in the sense that I haven't ...
1
vote
1answer
23 views

Proving path length, transitive closure

Set A is finite with n elements. Suppose a and b are elements of a set A with a != b. Let R be a relation on the set A so that there is a path from a to b of length at least 1. Show there is a path ...
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votes
1answer
38 views

Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
2
votes
2answers
29 views

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field?

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field ? I am claim that it is not. My reasoning is this... What I am ...
0
votes
1answer
23 views

Is this a valid proof of set membership

Let $S=\{x \in\mathbb{Z}: x\geq0, x=b-a ×m$ for some $m\in\mathbb{Z}\}$. Prove that if $b\geq0$ then $b$ is an element of $S$. Pf: suppose $b\geq 0$ Let $a$ be an integer define $b=b-a×m$ Where ...
2
votes
2answers
66 views

Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
2
votes
2answers
31 views

On the limit of $f(n)$, specifically having to do with integration of an iterated $\arctan$

Assume we are given that $A_n(x)$ denotes $n$ iterations of $\arctan(x)$, for example $A_2(x)=\arctan (\arctan(x))$ If $$f(n)=\int_{0}^n A_n(x)\space \text{d}x$$ I am looking for a rigorous proof ...
2
votes
2answers
58 views

Is the writing of the proof ok?

Problem. Let $f:(0,\infty)\to\mathbb{R}$. Prove that, $$\displaystyle\lim_{x\to\infty}f(x)=L\iff\displaystyle\lim_{x\to0 +}f\left(\dfrac{1}{x}\right)=L$$ My Solution. Let us assume that $\...
0
votes
1answer
62 views

Generalized DeMorgan's Law proof

We wish to verify the generalized law of DeMorgan $(\bigcup_{i \in \mathcal{I}} A_i)^c = \bigcap_{i \in \mathcal{I}} A_i^c$. Let $ x \in (\bigcup_{i \in \mathcal{I}} A_i)^c$. Then $x \notin \...
0
votes
1answer
37 views

Linear Algebra Proof - Columns of Matrix Linearly Independent & Determinant

How can I prove that if the columns of matrix A are linearly independent, then det(A) does NOT equal zero? This is a question on my exam review and I have no idea how to go about proving this. Any ...
-1
votes
1answer
48 views

Decide whether logical formula is a tautology [duplicate]

How do we decide whether the formula in predicate logic is a tautology? Is there some universal way to decide and prove it? Let's have an example: $$ \forall x \forall z \exists y\,(P(x,y) \lor P(y,...
0
votes
2answers
103 views

How to check, whether the formula is a tautology

How do we decide whether the formula in predicate logic is a tautology? Is there some universal way to decide? Let's have an example: ...
1
vote
1answer
59 views

Proof: Fibonacci Sequence (2 parts)

Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers. I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely ...
0
votes
0answers
4 views

Approximating semicontinuous functions by continuous functions. [duplicate]

Let $f=f(x):[0,1]\to\mathbb{R}$ be a upper (or lower) semicontinuous function, i.e., $$\limsup_{j\to\infty}f(x_{j})\le f(x)\quad\text{for $x_{j}\stackrel{j\to\infty}{\longrightarrow}x$}$$ (or $\...
3
votes
0answers
40 views

Leibniz Notation for the Derivative of a Function

I am writing a professionally-written proof, and I have come across a bit of an issue regarding how to write the derivative of a function $H(t)$ with respect to $t$. Is $\frac{dH}{dt}$ an acceptable ...
2
votes
3answers
61 views

Proving $\sqrt{2}x-\sqrt{x^{2}+1} \geq \frac{\sqrt{2}}{2}\ln{(x)}$

How can I prove that $$ x\sqrt{2}-\sqrt{x^{2}+1} \geq \frac{\sqrt{2}}{2}\ln{(x)} $$ It's a derivation-based process if I remember correctly, however I was unable to prove it correctly.
1
vote
2answers
48 views

Prove that a sequence is bounded/unbounded

I'm trying to do a maths problem which requires me to determine whether a sequence is bounded or unbounded and then it wants me to prove my answe. I know that it's bounded but I've no idea how to ...
1
vote
1answer
24 views

How to define two functions in a clear and standard way?

I am working on a question and before I ask it, I wanted to get help in defining two functions clearly in a standard way. Here are the two functions: $f(a,x,p)$: count of the number of pairs $k,k+2$...
3
votes
7answers
216 views

Proving $\cos(x)^2+\sin(x)^2=1$

I need to prove that $\cos(x)^2+\sin(x)^2=1$ Here's how I started (using the Cauchy product): \begin{align} \cos(x)^2+\sin(x)^2 &=\sum_{k=0}^{\infty}\sum_{l=0}^k(-1)^l\frac{x^{2l}}{(2l)!}(-1)^{k-...
3
votes
1answer
43 views

$3x + 1$ is even iff $5x-2$ is odd

I'm asked to prove 'Let $x \in Z$. $3x + 1$ is even iff $5x-2$ is odd'. I have the following proof techniques in my toolbox: trivial/vacuous proofs (not so relevant in this case), direct proof and ...