Prove that a specific subset $A$ of a nontrivial vector space $V$ over an infinite field $\mathbb{F}$ is infinite

Question

Let $V$ be a nontrivial vector space over an infinite field $\mathbb{F}$. Suppose $V = \bigcup\limits_{i=1}^{n} S_i$, where $S_i$ is a proper subspace of $V$. We assume that $S_1$ is not included in $\bigcup\limits_{i=2}^{n} S_i$.

Let $w \in S_1 \setminus \bigcup\limits_{i=2}^{n} S_i$ and let $v \in V \setminus S_1$ Let $A = \{rw + v| r \in \mathbb{F} \}$.

I need to prove that $A$ is infinite.

$r_1w + v = r_2w + v \Rightarrow r_2^{-1}r_1w = w$. Not sure how to proceed.

Answer 1

I found what I've been missing. $aw = w \Leftrightarrow (a-1)w = 0 \Rightarrow a-1 = 0$ or $w = 0$. Since $w$ doesn't belong to every subspace of $V$, $w \neq 0$, hence $a-1 = 0$.

That said, $r_2^{-1}r_1 = 1$ and $r_2 = r_1$.

So, the fact that $A$ is infinite follows from the fact that $\mathbb{F}$ is.