An elevator ride down Steven stories

Find an equation of the tangent plane to the given surface at the specified point. $z=3y^2-2x^2+x,(2,-1,-3)$

Show that the least upper bound of a set of negative numbers cannot be positive.

A set of ordered pairs is called a _______.

Prove that for every natural number n, fraction $\frac{21n+4}{14n+3}$ is irreducible. I deduced that if we can prove that numerator and denominator have 1 as their GCD, we can get the result, but I cannot get it from thereon.

Let ${\left\{f_k\right\}}_{k=1}^{\mathrm{\infty }}$ be a sequence of $\mathcal{S}$-measurable real-valued functions on a measure space $(M,\mathcal{S})$. Then the functions
$\underset{k\to \mathrm{\infty }}{lim sup}{f}_k(x)\text{ and }\underset{k\to \mathrm{\infty }}{lim inf}{f}_k(x)$
are also $\mathcal{S}$-measurable.
I already have proven this statement and my question is about something else. In the script it says that the reader may use the hint:
For a sequence ${\left\{a_k\right\}}_{k=1}^{\mathrm{\infty }}$ of real numbers the following is true:
$\underset{k\to \mathrm{\infty }}{lim sup}{a}_k=\underset{k\to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}max\{a_k,a_{k+1},\dots ,a_m\}$
Now, I have not used this in my proof, but I can assure you, that my proof is airtight nonetheless. Anyway, can someone point out why this holds? I have not seen this equality before and did not know that one can present $lim sup,lim inf$ like that. I know that for a set of limit points $H$ we can say that $lim supH=maxH$ and vice versa. I do not even see how this property can be useful for our proof.

Simplify. $\frac{2n!}{(n-1)!n!4^n}$