Can a set (as a candidate to be a vector space) be shown to be closed under vector addition and scalar multiplication in one step?

Yes.

The latter two statements are specialisations of the first statement; the first statement being true implies that they are true, if you set $(\alpha = \beta = 1)$ and then $(\beta = 0)$ while $(\alpha)$ is allowed to take on an arbitrary value.

The truth of the first statement is implied from the latter two statements with "$(\alpha \textbf{v})$ and $(\beta \textbf{w})$ are in $V$" as an intermediate logical step.

Show that the minimal polynomial of $T:\mathbb{K}^n \mapsto \mathbb{K}^n$ remains the same over field extension

Proof of multiple equivalences.

$(\frac{a}{b}-1)^2+(\frac{b}{a}+1)^2\ge3$

Consider a set A = {0,1,2,3}, a relation R from set A = {(0,0),(1,1),(2,2),(3,3)}. Is relation R transitive? If yes, how?

Counterexample to the Converse of Baire's Category Theorem

Solving system of equations, (2 equations 3 variables)

Dimension of maximal totally isotropic space over finite field with standard bilinear form

Coin probability

Is this operator bounded and closed?

Integral operator $\frac{g(s)}{1+(\cdot-s)^2}$ on $L^2(\mathbb{R})$

Prove that $\sqrt{x+2} + \sqrt{2x-2} + \sqrt{9-3x} < 3\sqrt{3}$

Find the values, $p \in [1,\infty)$, such that the sequence $f_n(x) = \frac{n^2}{(n^2+x)\,x^{\frac{1}{3}}}$ is convergent in $L^p((1,\infty),\mu)$