Intereting Posts

Solutions for $ \frac{dy}{dx}=y $?
Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.
Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?
A question about cardinal numbers in ZF set theory.
Show that this piecewise function is differentiable at $0$
Sum involving the hypergeometric function, power and factorial functions
Kolmogorov-Smirnov two-sample test
Simplifying $\sum_{r = 0}^{n} {{n}\choose{r}}r^k(-1)^r$
Inscribed kissing circles in an equilateral triangle
If a derivative of a continuous function has a limit, must it agree with that limit?
Is there a relationship between trigonometric functions and their “co” functions?
Must a monotone function have a monotone derivative?
Why any short exact sequence of vector spaces may be seen as a direct sum?
Solving $x^2 \equiv 1 \pmod{p^{\ell}}$
Is there a simpler way to falsify this?

If $A$ is denumerable, is the set of all injective functions $A\to A$ equipotent with $2^A$?

I have proved $\aleph_0^{\aleph_0} = 2^{\aleph_0}$.

- prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$
- Is there a notation for being “a finite subset of”?
- Build a bijection $\mathbb{R} \to \mathbb{R}\setminus \mathbb{N}$
- How to define a well-order on $\mathbb R$?
- Cardinal equalities: $\aleph_0^\mathfrak c=2^\mathfrak c$
- Relation between XOR and Symmetric difference

- Preimage of generated $\sigma$-algebra
- Namesake of Cantor's diagonal argument
- Borel Measures: Atoms (Summary)
- How to define a bijection between $(0,1)$ and $(0,1]$?
- Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions
- set theory proof of $A\cap B = A \setminus(A\setminus B)$
- Study of Set theory: Book recommendations?
- Proving the inverse of a relation exists
- Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
- How does Cantor's diagonal argument work?

Let $I$ be the set of such injective functions. Partition $\mathbb{N}$ into countably many disjoint countably infinite sets $N_0,N_1,\ldots$. Every element of $\prod_n N_n$ is an injective function. Clearly, $$\prod_n\{0,1\}=2^\mathbb{N}\leq\prod_n N_n\leq I\leq\mathbb{N}^\mathbb{N}\leq 2^\mathbb{N}.$$ Since the ordering is antisymmetric and transitive, $|I|=|2^\mathbb{N}|$.

- If $x_n\leq y_n$ then $\lim x_n\leq \lim y_n$
- How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
- A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$
- Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$
- Expected number of triangles in a random graph of size $n$
- Best books on A Second Course in Linear Algebra
- Why is ZF favoured over NBG
- Find all the continuous functions such that $\int_{-1}^{1}f(x)x^ndx=0$.
- How can I visualize division of negative numbers
- Leinster question on isomorphic functor categories
- Homogeneous riemannian manifolds are complete. Trouble understanding proof.
- Pursuit Curve. Dog Chases Rabbit. Calculus 4.
- The limit of a sum (a Riemann sum?)
- Surface of genus $g$ does not retract to circle (Hatcher exercise)
- condition for cones to be reciprocal