If $X$ is a vector space with $\dim X =n $, and $E=(x_1, ..., x_n)$ is a set of $n$ vectors, $E$ spans $X$ if and only if $E$ is independent.
This is theorem $9.3 (a)$ from Rudin, Principles of Mathematical Analysis. I don't understand the beginning of the proof.
Rudin starts with this observation: since $\dim X =n $, we have that: $\forall y\in X, (x_1, ...,x_n, y)$ is dependent.
This observation didn't seem obvious to me, so I tried to prove it by contradiction:
Suppose $\exists y\in X $ such that $ (x_1,...,x_n, y )$ is independent. Then, as the set $ (x_1,...,x_n, y )$ is included in $X$, we would have that $\dim X \ge n+1$. Contradiction.
However, this little reasoning has led me to another thought: is $$\exists y \in X , (x_1,...,x_n, y ) \text{ independent} \tag{1}$$
equivalent to $$ \exists (x_1, ...,x_{n+1}) \text{ independent} \tag{2}$$
?
There is one way (the one that I used in my short proof above) that seems rather obvious (if $(x_1,...,x_n)$ are fixed and we append $y$ at the end then we have the existence of $(x_1, ...,x_{n+1})$).
Thank you very much in advance.