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.

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?

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 $\sum_{n=1}^{\infty}{x_n} < \infty$,$\sum_{n=1}^{\infty}{|y_n – y_{n+1}|} < \infty$. 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}| + […]

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 […]

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 $[a,b]$?

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 $\int_Ag(z)dz = 0$ […]

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. […]

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!

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.

Intereting Posts

List of interesting integrals for early calculus students
The algebraic closure of a finite field and its Galois group
Help understanding the difference between the LLNs and CLT?
What does “working mathematician” mean?
On the solution of constant coefficients PDEs (exponential method)
Abelian group admitting a surjective homomorphism onto an infinite cyclic group
variance of number of divisors
How many numbers are in the Fibonacci sequence
If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$
Problem on the number of generators of some ideals in $k$
Egyptian fraction representations of real numbers
Proving that a polynomial is irreducible over a field
Are there prime gaps of every size?
Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.
Teaching myself differential topology and differential geometry