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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1answer
55 views

How to interpret the logic of an “or” in a matrix proof.

I am trying to learn to better interpret the meaning of equations and that is the purpose of this question, not just to find the proof, but to find the logical flow of the proof and understand it. I ...
2
votes
2answers
33 views

Next step in proof of sets

Proposition to prove : (A-B)∩(B-A) = 0 So, I understand why this is 0, I'm just not sure what propositions should be used in proving so. I have this so far 1)(A-B)∩(B-A) :Premise ...
3
votes
1answer
28 views

If $H,K$ are subgroups of $G$, and $G$ is finite, prove that $[K\colon (H\cap K)]\leq [G\colon H]$

Let $H,K$ be subgroups of a finite group $G$. Prove that $[K\colon (H\cap K)]\leq [G\colon H]$. This is what I have: $[K\colon (H\cap K)] = |\left\{ a(H\cap K) \mid a\in K\right\}|$ $[G\colon H] = ...
0
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1answer
65 views

If a sequence ${a_n}$ is monotonically increasing. then $\lim_{n \to \infty} a_n = \sup{(a_n)}$

Can you please tell me if my proof is correct: If a sequence ${a_n}$ is monotonically increasing. Then $$\lim_{n \to \infty} a_n = \sup{(a_n)}$$ Proof: $$a_n\leq a_{n+1}\leq \sup(a_n)$$ Assume ...
0
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2answers
33 views

Proof by induction with variable other than $n$ [duplicate]

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
2
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1answer
75 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
2
votes
2answers
48 views

Show that $\int_{x=a}^{x=b} f'(x) g(x) dx=f(b)g(b)-f(a)g(a)-\int_{x=a}^{x=b} g'(x)f(x)\, dx$

I have to prove the following: Suppose $f$ and $g$ are differentiable on $[a,b]$ and $f'$ and $g'$ are integrable on $[a,b]$. Prove that $f'g$ and $g'f$ are integrable on $[a,b]$ and that of: $$ ...
1
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4answers
101 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
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2answers
84 views

Proving a complete and totally bounded metric space is compact.

I'm having trouble writing down the details of this proof formally. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, ...
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0answers
35 views

How to prove unicity in a disjunction of $n$ propositions

Let's suppose I have the propositions $\varphi_1, \varphi_2,...,\varphi_n$ and I want to prove that there happens exactly one of them. How do you do it? To do it the hard way I guess we first need to ...
0
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1answer
43 views

There exists a positive real number $u$ such that $u^3 = 3$

Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$. Show that there exists a positive real number $u$ such that $u^3 = 3$. So far, I have come up ...
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2answers
99 views

Proving logic statements

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Use this definition to prove or disprove the following with a structured proof technique: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. ...
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2answers
52 views

Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom

I have no idea whether this question is way too specific or whether something similar has already been asked (we still need to work out a way to search for formulas I guess). Anyways here I go: I ...
2
votes
0answers
18 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
0
votes
0answers
76 views

Justify each step in the following proof of Proposition 3.9 (b). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(b): If a ray emanates from an interior point of triangle ABC, then it intersects one of its sides. proof (a) Let r be a ray emanating from an interior point D. (b) The ray AD ...
1
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2answers
26 views

Prove that $(1+x)^n ≥1+nx$ for all $x>-1$ and $n=1,2,\ldots$

Prove that for every real number $x > −1$ and every $n = 1,2,\ldots,$ $$(1+x)^n ≥1+nx.$$ I don't know where to begin so I haven't tried anything.
0
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1answer
108 views

Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(a): If a ray r emanating from an exterior point of triangle ABC intersects side AB at a point between A and B, then r also intersects side AC or side BC. proof. (a) Let r= array XD ...
0
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1answer
16 views

How to prove a subset is over an interval using the definition of subset.

Determine whether {x ∈ R : x2 < 9} is a subset of the interval [0, 9]. Prove your answer using the definition of A is a subset of B.
7
votes
2answers
361 views

Proving a palindromic integer with an even number of digits is divisible by 11

I'm in an introductory course for discrete math so I'm a novice at English proofs. I'm not sure if my reasoning here is valid or if I'm using modular arithmetic correctly. Specifically the line I ...
0
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2answers
43 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
0
votes
1answer
42 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
0
votes
1answer
97 views

Limit Delta-Epsilon proof

Prove $\lim_{x \to a} 2x = 2a$ Using the formal proof, not informal. So we know $2|x - a| < \epsilon$ We need to find some $\delta$ We only need to prove there IS SOME $\delta$ right? Only ...
0
votes
4answers
94 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
0
votes
1answer
21 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
1
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3answers
51 views

Proof that when repeatedly splitting a heap of marbles into two and writing down the product of the two heap sizes, the total is ${x \choose 2}$

Here is the problem in full: A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
1
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1answer
58 views

Prove: $\{α_1,…,α_n\} ⊨ α$ iff $\{α_1,…,α_{n−1}\} ⊨ (α_n→α)$.

Recently began my second logic course and have been surprised at how very, very different it is from the first one. My main struggle is that we have to prove things all the time, and I've never learnt ...
3
votes
3answers
332 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...
0
votes
2answers
19 views

Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
0
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0answers
40 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
0
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2answers
59 views

If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
0
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1answer
19 views

Proving that the $k$th element of $A \cup B$ is median of (the first $k$ elements of A) $\cup$ (the first $k$ elements of $b$)

By union here, I am referring to a union where duplicates are allowed. Given two sorted arrays, A and B, how do you prove that the $k$th element in the union of A and B is the median of the following ...
0
votes
1answer
29 views

Prove that if a|c and b|c, and a and b are relatively prime, than ab|c

How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
4
votes
3answers
61 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
0
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3answers
40 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
1
vote
1answer
58 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
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2answers
82 views

Assume that 495 divides the integer 273x49y5 where x,y ∈ {0,1,2…9}. Find x and y.

So, I know that $495 = 5\times 9\times 11$. So then, if that's the case, then the number $\overline{273x49y5}$ must be divisible by $495$ if and only if it is divisible by 5 and 9 and 11. Then, I ...
0
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2answers
47 views

Let n ∈ ℕ. If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

I know how to test the divisibility of a number by 9, but only if I am given what n is. How would I set this problem up?
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1answer
62 views

Prove $2n+3 \le 2^n$ for all integers $n \ge 4$.

I have already started the problem but I am unsure on how to proceed. Prove $2n+3 \le 2^n$ for all integers $n \ge 4$. Base Case: Choose $n = 4$. $2n + 3 \le 2^n$ $2(4) + 3 \le 2^4$ $8 + 3 \le ...
-1
votes
2answers
41 views

Geometric summation proof, not calculus

I am trying to take the expression $$T=\sum_{k=1}^nkx^k$$ and make it into a "simpler expression." I have an example similar to it where i am finding $$\sum_{k=1}^nx^k$$ where the answer is $$S_0 = ...
1
vote
6answers
1k views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
0
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0answers
14 views

Proving relations are orders

The Problem Let P and Q be posets with respect to some order $\sqsubseteq$. Proof that the following relations are indeed orders. If P' is a subset of P, then P' is also a poset with ...
1
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1answer
206 views

Prove $\sum_{n=1}^\infty \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}$ converges and evaluate it.

Noting that the series inside is a part of the alternating harmonic series multiplied by $-n$, we get $\displaystyle \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}=\sum_{m=1} ^\infty \frac{(-1)^mn}{m} ...
1
vote
2answers
47 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
2
votes
2answers
72 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
1
vote
3answers
85 views

If $x^2$ is divisible by $4$ then $x$ is even?

I am studying discrete mathematics as course and I have to prove this "If $x^2$ is divisible by $4$ then $x$ is even". I am wondering how to prove it using the contrapositive of this ...
0
votes
2answers
29 views

Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
3
votes
2answers
81 views

Prove that a function is both differentiable and continous at a point $x_0$

Suppose $f$ is differentialble on $(a,b)$, except possibly at $x_0 \in (a,b)$ an is continous on $[a,b]$; assume $ \lim\limits_{x\rightarrow x_0}f´(x)$ exists. Prove that $f$ is differentiable at ...
1
vote
0answers
40 views

Prove that $f$ is uniformly continous

I have to prove this: Suppose $f:(a,b)\to \mathbb{R}$ is differentiable and $|f´(x)| \leq M$ for all $x\in (a,b)$.Prove that $f$ is uniformly continous on $(a,b)$.Give an example of a function ...
0
votes
1answer
19 views

Use the mean value to prove a certain result

I need to prove the following: Use the Mean-Value Theorem to prove that: $$\sqrt{1+h}<1+\frac{1}{2}h$$ for $h>0$ My attempt: we first note that given that $h>0$ then $$1+\frac{1}{2}h ...
1
vote
2answers
43 views

How to formalize proofs

I'm struggling a bit with my discrete maths course and I was wondering if anyone could help me with my assignment. The question I'm working on is, Prove that if a and b are positive integers, then ...