Intereting Posts

Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
The number of subspaces of a vector space forming direct sum with a given subpace
Proving Holder's inequality using Jensen's inequality
Isomorphism between $U_{10}$ , $U_8$ , $U_5$
Prove that $6|2n^3+3n^2+n$
Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$
Show that all groups of order 48 are solvable
Extending to a disc means fundamental group is trivial
Triangle Formula for alternative Points
Limit $\lim_{x\to 0} \frac{-\frac{2x}{e}}{\ln(1+(-\frac{2x}{e}))}$
Natual density inside a subsequence
About connected Lie Groups
A certain “harmonic” sum
A polynomial is zero if it zero on infinite subsets
Straight lines – product of slope of perpendicular lines.

To prove $\tan(x)$ defined on $]-π/2;π/2[$ is injective I take the derivative of $\tan(x)$ to get $\sec(x)^2$.

This shows that $\tan(x)$ is monotonic (strictly) increasing which implies it is injective.

However how do I show it is surjective ? That every single real number corresponds to some number in the domain of $\tan(x)$ ?

- Inverse of Heine–Cantor theorem
- Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$
- A question about using the squeeze theorem?
- trouble calculating sum of the series $ \sum\left(\frac{n^2}{2^n}\right) $
- Evaluate $\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$
- Convergence of $\int_{1}^{\infty}x\cdot \sin{(e^x})\,dx$

Thanks in advance

- Existence of mixed partials in Clairaut's theorem.
- Lipschitz in $\mathbb R^1$ implies Lipschitz along any line in $\mathbb R^k$ (for convex functions)
- Is $\sum_{n=1}^{\infty}\frac{(\log\log2)^n}{n!}>\frac{3}{5}$
- Bijection between an infinite set and its union of a countably infinite set
- $f \in {\mathscr R} \implies f $ has infinitely many points of continuity.
- Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials
- show that $ \lim(\sin(\frac{1}{x}))$ as $x$ approaches zero does not exist.
- Uniform Integrbility
- Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
- Continuity of $\delta$ in the definition of continuity

I can see that a rigorous analytical proof for the surjectivity of $f\colon(-\pi/2,\pi/2)\to\mathbb{R}$, where $f(x)=\tan(x)$, is a way far off. However, I wanted to give a picture of what @achillehui mentioned in a comment, as it is rather beautiful in my opinion:

$\color{white}{Put it in the center!!!!!!!}$

Given the picture above, we note that

$$

\tan(\theta)=\frac{x}{1}=x\qquad\text{and}\qquad\tan(-\theta)=-\frac{x}{1}=-x.

$$

Hence, we can “clearly” see that

$$

\lim_{\theta\to\pi/2^-}\tan(\theta)=\infty\qquad\text{and}\qquad\lim_{-\theta\to-\pi/2^+}\tan(-\theta)=-\infty.\qquad\approx\blacksquare

$$

- Thomson's Lamp and the possibility of supertasks
- Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
- Decomposition of a nonsquare affine matrix
- Can forcing push the continuum above a weakly inacessible cardinal?
- weak sequential continuity of linear operators
- Proving sequence equality using the binomial theorem
- $G$ group, $H \trianglelefteq G$, $\vert H \vert$ prime, then $H \leq Z(G)$
- Why hyperreal numbers are built so complicatedly?
- Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$
- Rank-one perturbation proof
- Homeomorphic or Homotopic
- Perfect squares formed by two perfect squares like $49$ and $169$.
- Linear dependence of linear functionals
- Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves
- A problem on functional equation $f(x+y)=f(x)f(y)$