# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

65 views

### Is it necessary that $A$ has $m-1$ eigenvectors corresponding to $\lambda$, which are orthogonal to vector of all ones

If $A$ is real symmetric matrix and has an eigenvalue $\lambda$ with multiplicity $m$, Is it necessary that $A$ has $m-1$ eigenvectors corresponding to $\lambda$, which are orthogonal to vector of all ...
63 views

### Theorem 4.6 from baby Rudin

Theorem 4.6. In the situation given in Definition 4.5, assume also that $p$ is a limit point of $E$. Then $f$ continuous at $p$ if and only if $\lim \limits_{x\to p}f(x)=f(p)$. Proof: Let $f$ ...
115 views

### Integrability of Thomae's Function on $[0,1]$.

Consider the function $f: [0,1] \to \mathbb{R}$ where f(x)= \begin{cases} \frac 1q & \text{if } x\in \mathbb{Q} \text{ and } x=\frac pq \text{ in lowest terms}\\ 0 & \text{otherwise} ...
27 views

60 views

### Prove $\arctan(x+y)<y+\arctan(x),x\in \mathbb{R}, y>0$

Prove $\arctan(x+y)<y+\arctan(x),x\in \mathbb{R}, y>0$ Using Lagrange's mean value theorem, $$f(u)=\arctan(u)$$ In the first case, $x>0$ $$x=a,x+y=b$$ $f(u)$ is continuous and ...
62 views

### Show that $\langle 5, x^2+x +1 \rangle$ is maximal ideal in $\mathbb{Z}[x]$.

Here is my try, of which I'm rather skeptical. Let $I$ an ideal such that $\langle 5, x^2+x +1 \rangle \subset I \subset \mathbb{Z}[x]$. Because of the containment, there must be some $\alpha \in I$ ...
67 views

### Is my induction proof of $2^{n} > 2n+1$ correct?

Hello I am wondering if anyone can conform that the method I use in the following proof is valid. If not please inform me/ point me in the right direction. It is a very basic question, i.e. to prove ...
69 views

### A question about conic section (ellipse).

I am asked to solve the problem what is the center of the ellipse with vertex $V_1=(1,3)$ and focus $F_1=(1,0)$ and eccentricity $e=1/2$. My answer is due to the following analysis and computation: ...
340 views

112 views

### Triangle Inequality Equality Conditions

I am looking for the conditions on two complex numbers $z_1$ and $z_2$ such that $$|z_1+z_2|=|z_1|+|z_2|$$ Letting $z_n=a_n+ib_n$ and using $|z_n|^2=a_n^2+b_n^2$ yields ...
350 views

### Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?

Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative? My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all ...
171 views

### Examples of monotone functions where “number” of points of discontinuity is infinite

We know that if $f:D(\subseteq \mathbb{R})\to\mathbb{R}$ be a monotone function and if $A$ be the set of points of discontinuity of $F$ then $\left\lvert A \right\rvert$ is countable. Where ...
60 views

### Trying to prove $\sum_{i=1}^{N} i^3 = (\sum_{i=1}^{N} i)^2$

I'm trying to prove $\sum_{i=1}^{N} i^3 = (\sum_{i=1}^{N} i)^2$ but I got stuck along the way. This is what I have so far: The base case is true when $N =1$. Then for the inductive step I did: ...
13 views

### PRNG Improvements

Purpose This is the (somewhat) mathematical representation of an algorithm for a pseudo random number generator. It uses mostly linear math and generally is not very complex, but then again - I'm not ...
63 views

### Identify the error - Discrete math

I'm having problems trying to identify the error in this proof in the question below: Let $u$, $m$, $n$ be three integers. If $u\mid mn$ and $\gcd(u,m) = 1$, then $m = \pm1$. If $\gcd(u,m) = 1$, ...
52 views

### Question about proof for why every partial order on a nonempty finite set has a minimal element

The proof goes as follows: Proof. Let $R$ be a partial order on a set, $A$. For any element, $a ∈ A$, let $g(a)$ be the set of elements “less than or equal to $a$”, that is, ...