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Let $V$ be a vector space (can be finite or infinite) over finite field $K$, such that $dim V > 1$ and $|K| = q < infty$. Prove there exist proper subspaces $V_0, dots, V_q$ such that $V = V_0 cup dots cup V_q$. I have no idea where to start from.

We claim that if $V$ is a vector space over a finite field $K$ of order $q$ such that $dim V>1$, and $m$ is a non-negative integer, then $V$ can be written as a union of $m$ proper subspaces of $V$ if and only if $mgeq q+1$. (From the proof below, it also follows that if $K$ is not finite, then there is no way to cover a vector space $V$ over $K$ with $dim V>1$ by finitely many proper subspaces.)

First suppose that $mgeq q+1$. It suffices to assume that $m=q+1$. Pick a basis $mathcal{B}$ of $V$. Let $a,binmathcal{B}$ be two distinct elements (noting that $|mathcal{B}|>1$ since $dim V>1$). For each $kin K$, we define $V_k$ to be the span of ${a+kb}cupbig(mathcal{B}setminus{a,b}big)$, and $U$ is the span of $mathcal{B}setminus{a}$. Show that
$V=Ucup bigcup_{kin K}V_k$.

Conversely, suppose that $V$ can be written as a union of $m$ proper subspaces $W_1,W_2,ldots,W_m$ with $m$ being smallest possible (from the previous paragraph we know $m$ exists, so taking the smallest one is possible). It is easy to see that $m>1$. By minimality of $m$, for any $i$, we have $W_inotsubseteq bigcup_{jneq i}W_j$.

Take $uin W_1setminusbigcup_{jneq1}W_j$ and $vin W_2setminusbigcup_{jneq 2}W_j$. Since $u+svin V$ for all $sin K$ such that $sneq 0$, we must have $u+svin W_j$ for some $j$. We claim that the assignment $sin Ksetminus{0}$ to the smallest $j$ such that $u+svin W_j$ is an injective function from $Ksetminus{0}$ to ${3,4,ldots,m}$. From here, it follows that $$q-1=big|Ksetminus{0}big|leq big|{3,4,ldots,m}big|=m-2,$$
establishing our claim.

Now, to prove the assertion in the previous paragraph, we first note that $u+svnotin W_1$ and $u+svnotin W_2$ for $sne 0$. If $u+svin W_1$, then $v=s^{-1}big((u+sv)-ubig)in W_1$ since $uin W_1$, which is a contradiction. If $u+svin W_2$, then $u=(u+sv)-svin W_2$ since $vin W_2$, which is also a contradiction. So, $u+svin W_j$ for some $jin{3,4,ldots,m}$.

Now, suppose that there are two non-zero $s,tin K$ such that $u+sv$ and $u+tv$ are in the same $W_i$, where $iin{3,4,ldots,m}$. Then,
$$v=(s-t)^{-1}big((u+sv)-(u+tv)big)in W_i.$$
But $vin W_2setminus bigcup_{jneq 2}W_j$, so we have another contradiction. The assertion is now proven.

Hint: If $(x_1,x_2,ldots)in V$ then either $x_1=0$ or there exists $cin K$ with $x_2=cx_1$.