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

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2answers
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Prime numbers proof

My problem: Prove that a natural number $p$ is prime if and only if $p > 1$ and there exists no natural number $n \in \mathbb{N}$ with $1<n\le \sqrt{p}$ such that $n|p$. Help!
0
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2answers
44 views

How to prove $\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$ combinatorially

How can we prove combinatorially $$\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$$ I can get LHS by asking: How many ways can we form an $m+1$ person committee from a group of $n+1$ ...
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1answer
58 views

Choosing a $k$ person committee with chairperson from a group of $n$ people confusion

The following is from: http://www.math.sjsu.edu/~bremer/Teaching/Math163/Homework/HomeworkFiles/Solution03.pdf I am having trouble understanding these identities and the solutions. I am confused as ...
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0answers
23 views

Domain specification of derivative extension.

Given the definition of Taylor expansion: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ We can find the $m$'th derivative of $f(x)$ quite easily: $$\frac{d^m}{dx^m} f(x) = ...
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1answer
30 views

Proof based on convergence arguments that, if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \ | \ \phi (x) \leq \alpha \}$ is closed

Recently, I posted a proof of the proposition that, given a continuous function $\phi \in \mathbb{R}^X$, the set $\{ \ x \ | \ \phi (x) \geq \alpha \}$ is closed. Apparently, assuming that lack of ...
3
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4answers
76 views

How to prove if $m$, $m + 2$, $m + 4$ are all primes, then $m = 3$

I feel as though I have to use mods, but I'm not sure how exactly to go about this one.
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2answers
103 views

Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.

Recently, having realized I did not properly internalize it (shame on me!), I went back to the definition of continuity in metric spaces and I found a proposition for which I was looking for a proof. ...
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2answers
42 views

proof for $(1 + i)^n =(\sqrt{2})^n(\cos(\frac{n\pi}4) + i\sin(\frac{n\pi}4))$

How would I prove $(1 + i)^n =(\sqrt{2})^n(\cos(\pi n/4) + i\sin(\pi n/4))$ for all positive integers $n$
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2answers
40 views

For some $\beta,\rho \in S_X$, $\beta$ is an $r$-cycle, then $\rho\circ\beta\circ\rho^{-1}$ is also an $r$-cycle?

Some facts that I know: If $\beta$ and $\rho$ are disjoint, then $\beta$ and $\rho^{-1}$ are disjoint. If $\beta$ and $\rho$ are disjoint, then it is easy to show the required. If $\beta$ and $\rho$ ...
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1answer
34 views

Mathematical induction proving formula with only k-2

The sequence is defined by sk = 2sk-2, and the first 2 terms are 1 and 2. So the sequence looks like 1,2,2,4,4,8,8,16,16... I'm supposed to write an explicit formula for the sequence. I went with ...
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1answer
30 views

Deriving Preference Relations

I have actors A and B who have a preference relations $\succeq_A$ and $\succeq_B$ on a set $X$. Both are complete and transitive. Actor A will report Actor B's preferences as his own if Actor B ...
2
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2answers
98 views

Writing proof (disproof) prime number formula

Please prove or disprove: If $n \in ℤ^{+}$, then $n^{2} + n + 41$ is prime. I know that the above statement is not true because if you plug in 41 for $n$, the result is not a prime number. How can I ...
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1answer
42 views

Question from Real Analysis — making proof cleaner

I'm taking my first real analysis class after about maybe 7 years from the last time I took one. I have a question about how to make my proof nicer. The question is: Let $f : X \rightarrow Y$ be a ...
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3answers
66 views

How to prove that the square root of “$9n+3$” is not natural? [closed]

How should I prove that the square root of $9n+3$ is not natural?
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0answers
33 views

How to Prove It,4.4,Ex 23, Partial Orders

Theorem Suppose $A$ is a set, $F ⊆ P (A)$, and $F \ne ∅$. Then the least upper bound of $F$ (in the subset partial order) is $∪F$ and the greatest lower bound of $F$ is $∩F$. Proof: Since any element ...
3
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2answers
168 views

How can I prove $\lim \limits_{n \to \infty}\left(1+\frac{1}{a_n}\right)^{a_n}=e$ without involving function limit?

If I already know that $$\lim \limits_{n \to \infty} a_n=+\infty$$ Then how can I prove $$\lim \limits_{n \to \infty}\left(1+\frac{1}{a_n}\right)^{a_n}=e$$ without involving function limit? This ...
2
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2answers
51 views

Prove that using induction that $\binom22+\dots+\binom n2 = \binom{n+1}2$ [duplicate]

so I have this math problem where I have to prove this using induction. ...
1
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2answers
77 views

Prove that $(a+1)(a+2)(a+3)\cdots(a+n)$ is divisible by $n!$

so I have this math problem, I have to prove that $$(a+1)(a+2)(a+3)\cdots(a+n)\text{ is divisible by }n!$$ I'm not sure how to start this problem... I completely lost. Here's what I know: ...
2
votes
1answer
109 views

Proof using binomial Theorem

so I have to prove this using the binomial theorem: $$\sum_{k=0}^n{(-1)^k\begin{pmatrix}n\\k\end{pmatrix}}=0$$ I know the binomial theorem states: ...
2
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1answer
48 views

End of step symbol

This is more of a style question. We all know to end a proof with the good old QED (I use LaTeX's $\qed$ $\square$). I have a proof that is kind of long, and I was ...
0
votes
3answers
141 views

Prove the inequality for all natural numbers n using induction

$\log_2 n<n$ I know how to prove the base case Base Case $\log_2 1<1$ likewise assuming the inequality for n=k; $\log_2 k<k$ Then to prove by induction I show $\log_2 k<(k+1)$? I know ...
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2answers
212 views

Using Squeeze Theorem to prove two sequences converge to same limit

In a Real Analysis book without solutions, I came across the question Let $(a_{n})_{n=1}^{\infty}$ and $(b_{n})_{n=1}^{\infty}$ be two sequences of real numbers such that $|a_{n} - b_{n}| < ...
3
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2answers
81 views

Prove that $\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$ for all $n\geq1$

Prove that $$\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$$ for all $n\geq1$ I am looking for a clear solution to this problem. I've considering trying to prove it by contradiction by starting off ...
3
votes
2answers
66 views

What is wrong with representing an arbitrary natural and odd square number as $(2n-1)^2$?

Could I not represent every odd square number in $\mathbb N$ using the following notation: $(2n-1)^2$ where $n \in \mathbb N$. For every $n= 1,2,3...$ I get the set $1,9,25...$ every odd square ...
4
votes
1answer
81 views

Uniform Convergence of Maximum of Sequence of Functions

Let $K$ be a compact metric space, and $\{f_n\}_{n \in \mathbb{N}}$ is a uniformly bounded, equicontinuous family of functions. Define $$g_n(x) = \max \{f_1(x),f_2(x),\ldots,f_n(x)\}.$$ Prove that ...
0
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1answer
29 views

How to argue that $\int_{0}^{t} |x(s) - y(s)|dt \leq \delta \max\limits_{s \in [0,1]} |x(s) - y(s)|$

$\int_{0}^{t} |x(s) - y(s)| dt \leq \delta \max\limits_{s \in [0,1]} |x(s) - y(s)|$ was in the last step (on pg 12, http://faculty.cord.edu/obihun/BanachTalk1.pdf) to prove uniqueness of ODE, where ...
0
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2answers
43 views

Any nonempty closed bounded set contains its LUB and GLB.

Although this seems intuitive, I don't quite see how to prove this. A set $A$ is closed provided if $a_n \in A$ with $a_n \to p$, then $p \in A$. Since $A$ is bounded, then any nonempty $a_n \in A$ ...
2
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2answers
55 views

Is this a viable proof?

I need to show that if $3n-1$ is odd then $n$ must be even. I'm doing this in cases. For the first case I am saying: $$n = 2k \Rightarrow 3n-1 = 6k-1$$ Let $$j = 3k \Rightarrow (3n-1) = (6k-1) = (2j ...
0
votes
6answers
86 views

If $a,b> 0$, $a\neq b$, and $a + b = 2$ prove that $ab < 2$.

Let $a$ and $b$ be two positive real numbers such that $a \neq b$. Also, $a+b=2$. Now it is required to prove that $ab<2$. Thanks for any systematic and mathematical proof.
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2answers
32 views

Proof by induction: $(1+\alpha)^n\ge 1+n\alpha > +\frac{n(n-1)}{2}\alpha ^2$

so I have this problem. It asks me to prove an expression by induction. Let $n$ be a positive integer, and $\alpha$ any nonnegative real number. Prove by induction that$$(1+\alpha)^n\ge ...
2
votes
3answers
73 views

Prove $\frac{(2n)!}{2^nn!}$ is always an integer by induction.

Hey guys so I have this math question. I have to prove that $\frac{(2n)!}{2^nn!}$ is always an integer by induction where $n$ is a positive integer. This is my approach. First I check the initial case ...
1
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1answer
58 views

Using Combinatorial proof to conclude $\binom{2n}{n} = 2 \binom{2n-1}{n-1}$

The question given states: Let x be an element of a set A of size 2n. Among the n-element subsets of A count those containing x and those omitting x. Conclude that $\binom{2n}{n} = 2 ...
1
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1answer
40 views

Prove A bijection is increasing on it's inverse as well as the original function

Let $f:A \rightarrow B$ be a bijection, where $A$ and $B$ are subsets of $\mathbb{R}$. Prove that if $f$ is increasing on $A$, then $f^{-1}$ is increasing on $B$. I have an idea of the picture of how ...
0
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1answer
44 views

Using the distributive law in propositional calculus

So I was given this proof in class: ~p ^ q = (~ p v q) ^ ~(~q ^ p) = (~ p v q) ^ (q v ~p) by double negative law. = ~p ^ (q ^ ~p) v ( q ^ ( q ^ ~p)) by distributive law. ...
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1answer
29 views

How to make the argument $\lim\limits_{k \to \infty} (a + b^k) = a, 0< b < 1$ for matrices

Suppose $a,b \in \mathbb{R}$, then we can easily see that $\lim\limits_{k \to \infty} (a + b^k) = a, 0< b < 1$ Suppose we have $A,B \in \mathbb{R}^{n \times n}$, $\|B\| < 1$ and attempt to ...
3
votes
3answers
54 views

Let $a, n \in \mathbb{Z}_{\geq 0}.$ Prove that the product $(a+1) \cdots (a+n)$ is divisible by $n!$

Let $a, n \in \mathbb{Z}_{\geq 0}.$ Prove that the product $(a+1)\cdots(a+n)$ is divisible by $n!$. I think that can be done using the rule that ${a+n \choose n}= \dfrac {(a+n)!} {(a+n-n)! (n)!} $, ...
0
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1answer
110 views

Proof: $f$ is continuous if and only if the preimage of every open subset $V$ of $Y$ is open.

I know there is a very similar question being asked here, but the answers to that question do not answer the question I have here. They use a different proof, and the one below is one my professor ...
0
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2answers
25 views

Is the following a subspace of $P_3$?

I am struggling with this question. I know that I need to use the subspace test but I am stuck on how exactly to do so for this example: Is the following a subspace of $P_3$? $$U = \{xg(x) + ...
0
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2answers
82 views

Discrete Math Proof; Find proof or counterexample

My intro to discrete math class homework is asking me to either prove or find a counterexample to the following statement: For any integer $n \ge 3$, the number $n^2 − 1$ is composite. I'm supposed ...
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1answer
42 views

Negation of a conjuntion in a actual proof.

I proved that if $Y$ is a proper subspace of a Banach space then interior of $Y$ is empty. But, looking at what I logically did, made me confused. To me it looks like I'm proving that $$(A \wedge B) ...
3
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2answers
68 views

Discrete math combinatorial proof

Prove that for every positive integer n, $$\sum_{k=0}^{n}\binom{n}{k}^2=\sum_{l=0}^{n}\sum_{r=0}^{\frac{n-l}{2}}\binom{n}{l}\binom{n-l}{r}\binom{n-l-r}{r}$$ I know that ...
1
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0answers
31 views

Show that $\mathcal{I}$ must contain at least one positive integer.

I proved part (ii) but when I look at part(i) I feel like I have shown that when I was proving Part (2). I wanted to get some feedback on this question. Let $\mathcal{I}$ be a non-zero ideal of ...
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1answer
34 views

How to prove $f(n)=\lceil\frac{n}{2}\rceil$ is one-to-one and onto?

Edit: Both domain and codomain are the set of all integers ($\mathbb{Z}$) I proved $f(n) = \lceil\frac{n}{2}\rceil$ was one-to-one this way: $\forall{x_1}\forall{x_2}[(f(x_1) = f(x_2)) \to(x_1 = ...
0
votes
1answer
22 views

Proving with division algorithm

$$ Let \ b \ be \ a \ natural \ number \ and \ q_1, \ q_2, \ r_1, \ r_2 \ integers \ with \ 0 \le r_1 \lt b \ and \ 0 \le r_2 \lt b \ such \ that \ q_1b + r_1 = q_2b + r_2 \ then \ q_1 = q_2 \ and \ ...
-1
votes
2answers
135 views

Let a and b be real numbers such that 0 < a < b. Prove $\frac{a+b}2 > \sqrt{ab} > \frac{2ab}{a + b}$

Let a and b be real numbers such that 0 < a < b. Prove $\frac{a+b}2 > \sqrt{ab} > \frac{2ab}{a + b}$ How can I prove this? Been working for hours and got nowhere. I see $\frac{a+b}{2}$ ...
0
votes
1answer
76 views

Prove that if $n^6$ is a perfect square, $n^{50}$ is a perfect square. [closed]

Can someone help me prove/disprove this? I wrote $n^6$ as $l^2$, but I don't know how to convert $n^{50}$ into that format because $^{50}$ is too large. $n$ ∈ ℤ
1
vote
2answers
39 views

Proving with well-ordering principle

I have this conjecture: Let a and b be integers and n and m natural numbers. $$ a \equiv b \bmod n \Rightarrow a^m \equiv b^m \bmod n$$ I think I got the induction proof, but I'm having ...
1
vote
3answers
42 views

How to write down this simple proof? (in natural numbers, if for every number there is a smaller number then 1 is in the set)

This seems undeniably true to me, but I don't know how to write it down. Given the non-empty set $S$ containing only natural numbers (starting at 1, not 0). If for every number $x$ greater than 1 ...
0
votes
1answer
55 views

A simple(ish) proof for the lagrangian with one inequality constraint?

I know the Lagrangian comes indirectly from the implicit function theorem (so don't worry about that nightmare) but does anyone know a good proof for the following theorem: Consider the optimization ...
1
vote
2answers
139 views

Proving with prime

Definition of prime is that a natural number $n > 1$ is prime if the only natural numbers $m$ with $m|n$ are $m = 1$ and $m = n$. I'm guessing this means that the prime numbers can only be divided ...