Very affine varieties form basis of Zarisky Topology

I am currently working through a paper (related to tropical geometry, but this is not important in the following context) which utilizes the concept of very affine varieties in the following way (I'll give a quick summary of the concept):

Let $$X$$ be a variety over a field $$K$$, i.e. an irreducible, reduced, separated $$K$$-scheme of finite type. Then an open subset $$U$$ of $$X$$ is called very affine if $$U$$ has a closed immersion to some multiplicative split torus $$\mathbb G^r_m := Spec(K[T_1^{\pm 1}, \dots, T_r^{\pm 1}])$$.

The author states: "[...] The very affine open subsets of $$X$$ form a basis for the Zariski topology."

I am wondering why this is the case. Here is what I know so far, if any of it helps: First, let $$U$$ be any open affine subset of $$X$$. Then $$\begin{equation} \text{Hom}_{K-\text{Sch.}}(U, \mathbb G^r_m) \cong \text{Hom}_{K-\text{alg.}}(K[T_1^{\pm 1}, \dots, T_r^{\pm 1}], \mathcal O_X(U)). \end{equation}$$

Any $$K$$-algebra morphism $$f \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U)$$ is uniquely given by the units $$f(T_i) \in \mathcal O_X(U)^*$$.
Furthermore it is known that the abelian group $$\mathcal O_X(U)^* / K^*$$ is free of finite rank, say $$r$$. Choose representatives $$\varphi_1, \dots, \varphi_r$$ in $$\mathcal O_X(U)^*$$ of a basis, we get a (more or less canonical) morphism $$\varphi \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U), T_i \mapsto \varphi_i.$$

So far so good. Using these identities it is easy to show that the following statements are equivalent:

a.) $$U$$ is very affine;
b.) $$\mathcal O_X(U)$$ is generated as a $$K$$-algebra by $$\mathcal O_X(U)^*$$;
c.) the canonical map $$\varphi$$ above is surjective.

Also I have shown that intersections of very affine opens are very affine again.

My approach now would be to take any point $$x \in X$$, choose any affine open $$U$$ around it and then try to 'make it very affine' by successively localizing until I get an open subscheme $$V$$ of $$U$$ that satisfies statement b.) above. However I can't really get it to work.

Long explanation, but if anyone could help or has any comments, that would be super awesome!

Thanks a lot in advance!

Claim: Let $$X$$ be variety over field $$K$$. Then the very affine open subsets of $$X$$ form a basis for the Zariski topology.
Proof: Let $$x \in X$$ and $$U \subseteq X$$ be an open neighborhood of $$x$$. It suffices to show that there is very affine $$V$$ around $$x$$ with $$V \subseteq U$$.
As open subschemes of varieties are varieties again, and by possibly passing to a smaller open neighborhood, we can assume that $$U$$ is affine with $$U = \text{Spec}(A)$$, where $$A$$ is a $$K$$-algebra of finite type, i.e. it is of the form $$A = K[T_1, \dots, K_n]/I$$ for some ideal $$I$$. Let $$\mathfrak p$$ denote the prime ideal of $$A$$ corresponding to $$x$$. Let $$\overline{T_i}$$ denote the class in $$A$$ of $$T_i \in K[T_1, \dots, K_n]$$.
We first suppose that there is no $$i \in \{1, \dots, n\}$$ with $$\overline{T_i} \in \mathfrak p$$. Then $$V := D(\overline{T_1}) \cap \dots \cap D(\overline{T_n}) = D(\overline{T_1 \cdot \cdots \cdot T_n}) \subseteq U = \text{Spec}(A)$$ is open around $$x$$ and corresponds to localization $$A[\frac{1}{\overline{T_1 \cdot \cdots \cdot T_n}}] = A[\frac{1}{\overline{T_1}}]\cdots[\frac{1}{\overline{T_n}}] =: B$$.
As $$\overline{T_i} \in B^* \ \forall i \in \{1, \dots, n\}$$ we obtain a surjective $$K$$-algebra morphism $$K[T^{\pm 1}_1, \dots, T^{\pm 1}_n] \to B, T_i \mapsto \overline{T_i},$$ which shows that $$V$$ is very affine.
On the other hand, suppose $$\exists i$$ with $$\overline{T_i} \in \mathfrak p$$. Denote $$J \subset \{1, \dots, n\}$$ the set of such indices. For any $$j \in J$$ we then must have $$\overline{T_j} + 1 \notin \mathfrak p$$ and we can proceed as above by successively localizing to obtain $$V$$ (however instead of localizing at $$\overline{T_j}$$ we use $$\overline{T_j} + 1$$ for all $$j \in J$$). Then again obtain surjective $$K$$-algebra morphism $$K[T^{\pm 1}_1, \dots, T^{\pm 1}_n] \to B, T_i \mapsto \begin{cases}\overline{T_i}\ \ \text{for } i \notin J \\ \overline{T_i} + 1 \ \ \text{for } i \in J \\ \end{cases},$$ which shows the claim.