Articles of real analysis

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 […]

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}$

for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}, b_{n+1} = \frac{a_n + b_n}{2}$. Show that $a_n, b_n$ converge to a common limit. Hint use inequality: $\sqrt{ab} \leq \frac{a+b}{2}$ attempt. Suppose $a_n\longrightarrow L_1$ and $b_n \longrightarrow L_2$ using the hint: $$\lim_{n \to \infty} 4a_nb_n < \lim_{n \to \infty} (a_n + […]

Variant of Cauchy Functional Equation

Consider the equation $$f(kx-f(x))=x=kf(x)-f(f(x))$$ for montonic $f$. What can we say about the solutions to this equation. Comparing with Cauchy equation $f(x+y)=f(x)+f(y)$, I think the solution must be somewhat close to being linear. Any hints. Thanks beforehand.

derivative bounded by a constant multiple of the function

$f$ is differentiable on $[a,b]$, $f'(x) \leq A|f(x)|$ where $A$ is a non-negative constant. If $f(a)=0$ show $f(x)=0, \forall x\in [a,b]$ I imagine the proof uses the Mean Value Theorem but I have not been able to get it to work. I know $|f(x)|=|f'(c)|(x-a)$ where $c \in [a,x]$, so $|f(x)| \leq A\ |f(c)|(x-a)$ where $c\leq […]

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k \hookrightarrow \mathbb{R}^n$ we have $\mu_{k}(U \cap i(\mathbb{R^k}))=0$. Does this imply $\mu_n(U)=0$?

A semicontinuous function discontinuous at an uncountable number of points?

This is an exercise from A second course on real functions. Do you have an example of a semicontinuous function defined on $[0,1]$ which is discontinuous at an uncountable number of points?