Articles of real analysis

Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition

Let $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ be a subset of $\mathbb{R}$. I have to prove using the open cover definition that this is compact. Could you help me, please?

Theorem 6.12 (b) in Baby Rudin: If $f_1 \leq f_2$ on $$, then $\int_a^b f_1 d\alpha \leq \int_a^b f_2 d\alpha$

Suppose $f_1$ and $f_2$ are Riemann-integrable with respect to $\alpha$ over $[a, b]$. If $f_1(x) \leq f_2(x)$ on $[a, b]$, then $$ \int_a^b f_1 d \alpha \leq \int_a^b f_2 d \alpha. $$ This is (essentially) Theorem 6.12 (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Here is my proof: As […]

Showing that a function is in $L^1$

I need to prove the following statement or find a counter-example: Let $u\in L^1\cap C^2$ with $u”\in L^1$. Then $u’\in L^1$. Unfortunately, I have no idea how to prove or disprove it, since the $|\bullet|$ in the definition of $L^1$ is giving me huge problems. I found counter-examples if either $u\notin L^1$ or $u”\notin L^1$, […]

Proof that a function with a countable set of discontinuities is Riemann integrable without the notion of measure

Let $f:[a,b]\to \mathbb{R}$ be a bounded function and $A$ be the set of its discontinuities. I am asking for a (direct) proof that if $A$ is countable then $f$ is Riemann integrable in $[a,b]$ that doesn’t explicitely, or implicitly, require the notion of sets of measure $0$ ( and of course without the use of […]

Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.

How would you prove or disprove that the function given by $$f(x,y) = \begin{cases} \dfrac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ is continuous at $(0,0$)?

Example of dynamical system where: $NW(f) \not\subset \overline{R(f)}$

Can you provide me an example of a Dynamical System continuous or discrete in which: $$NW(f) \not\subset \overline{R(f)}$$ $NW(f)$ is the set of non wandering points, i.e. all $x$ such that $\forall$ U open containing $x$ and $\forall$ $N>0$ there exists some $n>N$ such that $f^n(U) \cap U \ne \emptyset$. $R(f)$ is the set of […]

If $b_n$ is a bounded sequence and $\lim a_n = 0$, show that $\lim(a_nb_n) = 0$

This is a real-analysis homework question so I of course have to be very precise and justify anything or any theorem I use. If $b_n$ is a bounded sequence and $\lim(a_n) = 0$, show that $\lim(a_nb_n) = 0$ Intuitively, since $b_n$ is bounded, then sup($b_n$) is some finite number and therefore we can take an […]

Proving a function is continuous on all irrational numbers

Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb R\;\text{by}\;\displaystyle f(x) = \sum_{r_n \leq x} 1/2^n,\;x\in \mathbb R.$$ Prove that $f$ is continuous at each point of $\mathbb Q^c$ and discontinuous at each point of $\mathbb […]

$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$

Let $f\in C^2(\mathbb [0,1],\mathbb [0,1])$ such that $f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ Prove that $\max_{[0,1]}|f”|\geq 4$ Progress Applying Cauchy mean value theorem three times proves the existence of $\xi\in (0,1)$ such that $f'(\xi)=1$ $\eta\in(\xi,1)$ such that $\displaystyle f”(\eta)=\frac{1}{\xi-1} <0$ $\beta\in(0,\xi)$ such that $\displaystyle f”(\beta)=\frac{1}{\xi}>0$ If $\displaystyle \xi\leq \frac{1}{4}$ or $\displaystyle \xi\geq \frac{3}{4}$, we’re done. What about other […]

Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{{\pi}/{2}} \log(1+\cos x)\ dx$ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be helpful: $\displaystyle \int_0^{{\pi}/{2}} \log(\cos x)\ dx =-\frac{\pi}{2} \log2 $.