Question regarding the proof that every hamel basis of an infinite space is uncountable

Question

I am reading the following question:

Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

And I am wondering why

$$X=\bigcup_{n\in \mathbb N}X_n$$

Since the right hand side is just an union of some sets and the left hand side is the complete space. Shouldn't it be:

$$X=lin(\bigcup_{n\in \mathbb N}X_n)$$ ?

Answer 1

The notation $[v_1,\ldots,v_n]$ refers to the vector space spanned by $B=\{v_1,\ldots,v_n\}$ and not $B$ itself. If $x\in X$ then by definition of a Hamel basis, $x=\sum_{j=1}^ka_jv_j$ for some $k\ge1$, $a_j\in\mathbb{K}$. This implies $x\in X_k\subseteq\bigcup_{n\in\mathbb{N}}X_n$.