Articles of real analysis

Infinite set and countable subsets

Prove that a set $A$ is infinite if and only if $A$ contains a countable subset $C$. I know I have to build a sequence and then I’ll get a countable subset, but I don’t know how to build that sequence from a infinite set.

Projection of a set $G_\delta$

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps $G_\delta$ sets to Borel sets? i.e. If $A=\cap_n^\infty A_n$ with $A_n$ open sets, then $\pi(A)$ is Borel?

closed epigraphs equivalence

Is there a way to prove that the epigraph of any real function $f$ is closed iff $f$ is lower semi-continuous without using limit superior or inferior?

Suppose $\sum_{n=1}^{\infty}{x_n} < \infty$,$\sum_{n=1}^{\infty}{|y_n – y_{n+1}|} < \infty$

Suppose $\sum_{n=1}^{\infty}{x_n} < \infty$,$\sum_{n=1}^{\infty}{|y_n – y_{n+1}|} < \infty$. Then prove that $\sum_{n=1}^{\infty}{x_ny_n}$ converges. Since $\sum_{n=1}^{\infty}{x_n} < \infty$, by divergence test, we have $\lim_n{x_n}=0$ Since $ \sum_{n=1}^{\infty}{|y_n – y_{n+1}|} < \infty $, we have $ \sum_{n=1}^{\infty}{(y_n – y_{n+1})} < \infty $. By divergence test. $\lim_n{(y_n-y_{n+1})}=0$. Hence, we have $|y_n – y_m| \leq |y_n – y_{n-1}| + […]

Cesaro Means convergence

Show that if $\lim x_n =x$ with $x_n \in \mathbb{R}$ and $x \in \mathbb{R}$, then the sequence given by the averages $y_n = \frac{x_1 +x_2 + \dots + x_n}{n}$ also converges to $x$. Solution: Denote the limit of $(x_n)$ as $a$. Let $\varepsilon >0$ Then there exists a positive integer $N_1 > 0$ such that […]

Improper Riemann integral of bounded function is proper integral

Let $f:[a,b) \rightarrow \mathbb R$ be Riemann integrable on each compact subinterval of $[a,b)$ and bounded on $[a,b)$. Let $g:[a,b] \rightarrow \mathbb R$ be arbitrary extension $f$ ( i.e. $g|_{[a,b)}=f$). Why $g$ is Riemann integrable on $[a,b]$?

Primitive of holomorphic Function $\frac{1}{z}$ on an Annulus.

Given an Annulus with $A(0,r,R)$ show by considering Cauchy’s Theorem for primitives that there is no holomorphic function with $f'(z)=\dfrac{1}{z}$. I am struggling to picture this since but it seems like there are issues because $f(z)=\log z$ isn’t well defined in the same range as $\dfrac{1}{z}$. Am I looking to show that $\int_Ag(z)dz = 0$ […]

Inverse/Implicit Function Theorem Reasons?

I watched an ICTP lecture on elementary real analysis & the lecturer went to great pains to emphasize the importance of the intermediate value theorem because it is what generalizes to higher dimensions via connectedness & how Bolzano-Weierstrass generalizes to metric spaces & how just a few results following from these are what really matters. […]

Compute $\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$

How could the series below be computed ? $$\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$$ It’s not a series from a book, but a series I thought of many times, and I didn’t manage to figure out what I should do here. I’m just curious to know if there are some known ways for approaching such a series. Thanks!

How to show that $\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$

Show that: $$\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$$ All I’ve got so far is that the minimum of $x^x$ is $e^{-1/e}$. At this point I could compare $\pi/5$ to $e^{-1/e}$ but I’m required to prove both sides without using the calculator. This is all I’ve got at the moment.