# proving

1. ### strong maths induction proving

Let a0, a1, a2, . . . be the sequence defined recursively asa0 = 1, a1 = 2, ak = ak−1 + ak−2 for each integer k ≥ 2. Use strong mathematical induction to prove that an ≤ 2nfor each integer n ≥ 0. This is what i have so far with my working.Let P(n) be the statement that an≤2n Basic step: Prove...
2. ### Proving a Sequence's Convergence

Hello, can you please check my work? Prove sin(n^2)/n^(1/3) converges to 0 Proof for any e > 0 abs(sin(n^2)/n^(1/3)) < e sin(n^2)/n^(1/3) < e Since the range of sin(n^2) is [-1,1], sin(n^2)/n^(1/3) < or = 1/n^(1/3) so let 1/n^(1/3) < e. So 1/n < e^3 let N < or = n Then, sin(n^2)^3/n < or =...
3. ### Proving a line is perpendicular to another line in terms of A,B,C...

Hi guys, I'm really stumped with question 1 of this assignment and was looking for some help. All input would be appreciated. Cheers. Sent from my SM-G900I using Tapatalk
4. ### Proving a complex inequality

$$|e^z -1| \leq e^{|z|} -1 \leq |z|e^{|z|}$$ for all $z \in C$. I tried substituting $e^z = e^x (\cos y + i \sin y)$ and $|z| = \sqrt {x^2 + y^2}$ but after 2 pages of manipulations, I got back where I started, only messier forms. Any hints are appreciated.
5. ### Proving or disproving that X and Y are independent

I pretty much get the whole procedure. We compare f(x,y) with g(x)h(y). If they are equal then they are said to be statistically independent, otherwise they are not statistically independent. I have one question. Initially I plugged in 1 for x and 2 for y, and I concluded that x and y are not...
6. ### Proving a Function is Analytic

I would like to prove that this function is analytic. It is really hard for me to see. We suppose that f is analytic and zero free in a domain D, the function I wish to prove is analytic is: \int_{z_0}^z \frac{f'(\zeta )}{f(\zeta )} ~d\zeta My professor tells me there is a standard argument...
7. ### Proving aI+A is Positive Definite

>Let $A \in M_{n \times n}^\Bbb C$ be a self adjoint matrix. Prove that there is $a \in \Bbb R$ such that $aI+A$ is a positive definite matrix. What I did so far Let $v$ be a vector in an orthonormal basis of $A$ which exists since $A$ is self adjoint.$A$ is also diagonalizable so \$Av=\lambda...
8. ### Help with proving Young's Theorem and a derivative involving e

f(x1,x2,x3 )= x21e(3x2+x1x3)+(2x32/x1) Show that f12=f21 and f13=f31, which are implications of Young's Theorem. What I did was FOCs with regards to x1,x2,x3. So I ended up with; df/dx1= 2x1e(3x2+x1x3) -2x2^3/x1^2 df/dx2= 3x2e(3x2+x1x3) +6x2^2/x1 df/dx3=x31e(3x2+x1x3) ....After many...
9. ### Proving identities

Hey, I used to teach math ~15 years ago. Today a friend asked me to help her kid with some trig. I'm out of my depth! - and don't have the time right now to re-learn trig. There are two "Prove the identities" problems - below. Would great appreciate any help listing the steps involved...
10. ### Proving properties of convex hexagon

Suppose we have convex hexagon ABCDEF. All of its vertices are on the circle, while |AB| = |CD|. Lines AE and CF intersect at point G and lines BE and DF intersect at intersect at point H. Prove that lines GH AD and BC are parallel. I realised that AD and BC are parallel because ABCD is...
11. ### Proving the random walk??

Usually when one monitories a stock market, one wishes to find a trend in order to be able to predict the price and make some money in the process. My problem is the opposite. I have pattern that resembles a random walk, it is a set of values (40-50 values) measured at different time points...
12. ### Proving angle equivalency

Hi everyone, Is there a way to prove that angle B and angle C are equivalent in this situation? The argument my textbook made certainly seemed to imply that, and it really bothers me that I can't quite see it. I'm worried it's really simple and I'm overthinking it... I haven't taken geometry in...
13. ### Proving that matrix is regular.

If A, E are nxn matrices with all elements being real numbers and if (A+E)3=0, then prove that A is regular matrix. I know what it means that matrix is regular, it means that it's determinant is not zero so there exists A-1 and AA-1=E but i don't know how can i prove that all this works for...
14. ### How to describe a strategy on proving that A= ∅, etc.

6. Describe strategies to prove the following: (a)A ⊆ B. (b) A = ∅. (c) A = B. (d) The set A has one single element. I'm stuck on how to describe a strategy for (b) and (d). I think that for (d) I just have to prove that A contains ∅. But for (b) do I have to prove that A contains a single...
15. ### Transitive law in proving p → q ≡ ~q → ~p by deductive reasoning.

It's written in my book that in the solution of the example , the p → q ≡ ~q → ~p is proven by the Transitive Law. But nowhere in the solution can I find transitive law being used. What I know from the book is that by transitive Law (Trans.), (p→q)∧(q→r) ⇒ (p→r). Commutative law or p→q ≡...
16. ### Proving that |N|^ |N| <= |R|

Proving that |N|^ |N| = |R| I have already proved that if k1 = |A|, k2 = |B| , m = |C| and we know that k1 <= k2 then k1^m <= k2^m. I should use this to prove that |N|^|N| = |R|. So I proved that |N| ^ |N| <= |R|, how do I continue now?
17. ### proving a sequence is divergent

hi when trying to prove the following sequence is divergent (-1)^n+\frac{1}{n} using the first subsequence rule by showing that the sequence has two convergent subsequence's with different limits I have approached question as follows let the even subsequence be a2k (-1)^2^k+\frac{1}{2k}...
18. ### proving limit of (1-1/n)^n as n tends to infinity

Knowing that lim n->oo (1+1/n)^n = e, how can I prove that lim n->oo (1-1/n)^n = 1/e I've tried to use the lim n->oo f(n)/g(n) = A/B where f(n)=A and g(n)=B where in this case f(n) = (1+1/n)^n and g(n)=1/h(n) where h(n) = (1-1/n)^n but I get to a problem where I'd need to prove that lim n->oo...
19. ### Need assistance in proving this statement by mathematical induction

Hi guys. I'm not sure if this is the right thread to this question because my math background during my high school days really suck. Haha. Anyway, I'm tasked to show by mathematical induction that 4 exactly divides 5a -1, where a is a non-negative integer (a here is an exponent, not a variable...
20. ### Need help in proving that this statement is valid.. (for propositional logic)

Hi guys. Good day. I was tasked to prove that this is a valid statement using Chain of Reasoning, Proof by Contradiction, and Proof by Resolution: (Q ⇒ P) ⇒ P (Q ⇒ Q) ⇒ S (S ⇒ R) ⇒ ~(R ⇒ P) ∴ S However, I'm stuck at the part when I used dominance to simplify the equation. So what I did was...