Articles of real analysis

Density of positive multiples of an irrational number

Let $x$ be irrational. Use $\{r\}$ to denote the fractional part of $r$: $\{r\} = r – \lfloor r \rfloor$. I know how to prove that the following set is dense in $[0,1]$: $$\{\{nx\} : n \in \mathbb{Z}\}.$$ But what about $$\{\{nx\} : n \in \mathbb{N}\}?$$ Any proof that I’ve seen of the first one […]

Sequence of Rationals Converging to a Limit

I’m trying to show that for every real number $r$, there exists a sequence of rational numbers $\{q_{n}\}$ such that $q_{n} \rightarrow r$. Could I get some comments on my proof? I know that between 2 reals $r, b$ there exists a rational number $m$ such that $r < m < b$. So I can […]

Why the following function converges?

In the following link: Examples of dense sets in the complex plane we can see an example which says that the following function converges: $$f(x)=\displaystyle\sum_{n=0}^\infty\frac{1}{2^n}\frac{1}{x-r_n}\sin\left(\frac{1}{x-r_n}\right) $$ where $r_n$ is any fixed enumeration of the rational, and $f(r_n)=0$. Why this expression converges?

approximation of a continuous function by polynomials over a strictly continuous monotone function

Let $h:[0,1]\to \mathbb R$ be a continuous and strictly monotone function. Let $f:[0,1] \to \mathbb R$. Prove that there exists a sequence of polynomials $p_n :\mathbb R \to \mathbb R$ such that $p_n(h(x)) \to f(x)$. If I prove the case $h:[0,1]\to [0,1]$ with $h$ homeomorphism, then I’m done. I think that this case is easier, […]

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I’ll first define what I mean by a ‘component of a topological space’: For a topological space $X$, write $x\sim y$ if $\exists\ Y \subset X$ such that $Y$ is connected and $x, y \in Y$ (this is an equivalence relation. […]

Positive bounded operators

Let $A,B$ be positive self-adjoint bounded operators and $\lambda >0$ then I want to show that if $$A-B \ge 0 $$ that is $\langle x,(A-B)x \rangle \ge 0$ we have that the resolvents (whose existence is clear) satisfy $$(A+\lambda I)^{-1}-(B+\lambda I)^{-1} \le 0,$$ i.e. exactly the opposite relation. Although this is intuitively clear, I got […]

Improve my proof about this $C^\infty$ function even more!

In a sense, I did just ask this question. However, since the question is about how to write a beautiful looking proof and my proof is entirely rewritten, it seems like it should be a new question. What I’m looking for here are (aside from any actual mathematical problems in the proof of course) is […]

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2N}{N^{2}+n^{2}}\right)$$=$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right).1/N$. Now, replacing $1/N$ by $dx$, $n^{2}/N^{2}$ by $x^{2}$ and summation by integral, we have $$\lim_{N→∞}1/N\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right)= \lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=-N}\left(\frac{1}{1+n^{2}/N^{2}}\right)$$ $$=2\lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=0}\left(\frac{1}{1+n^{2}/N^{2}}\right)=?$$ I feel that I am very close to the final answer which is $2\int_{-1}^{1} dt/(1+t^{2})$. But I am stuck after this step. please complete […]

What is the reasoning behind why the ratio test works?

The ratio test says that if we have $$\sum_{n=1}^{\infty}a_n$$ such that $\lim_{n \to \infty} \dfrac{a_{n+1}}{a_n} = L$, then if: 1) $L < 1$, then $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, 2) $L > 1$, then $\sum_{n=1}^{\infty}a_n$ is divergent, and 3) $L = 1$, then the ratio test gives no information. I want to understand the mathematics behind […]

The sequence $a_n = \left(\frac{n}{n+1}\right)^{n+1}$ is increasing

I just wanna some help showing that the sequence $a_n = \left(\frac{n}{n+1}\right)^{n+1}$ is increasing, since it came up in some stuff I was doing and I’m not finding a quick solution. I actually just need it to be bounded below by a positive number, but I tested a bunch of values and it’s increasing so […]