For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

First show that $M_1\cup M_2$ is a subspace of $V$ if and only if $M_1\subseteq M_2$ or $M_2\subseteq M_1$.

This extends by induction to any finite number of subspaces: if $M=M_1\cup\ldots\cup M_k$ is a subspace of $V$ then there is some $i$ such that $M=M_i$.

Next show that if $N\subseteq M_1\cup\ldots\cup M_k$ then $N_i=N\cap M_i$ is a subspace of $V$, and therefore $N=N_1\cup\ldots\cup N_k$ and it is a subspace, so $N=N_i$ for some $i$, so $N\subseteq M_i$ as wanted.

Recall that a vector space over an infinite field is not a finite union of proper subspaces. Since your scalar field has characteristic $0$, it is necessarily infinite, as it contains an isomorphic copy of $\mathbb{Z}$ for instance.

If $N\subseteq M_1\cup\cdots\cup M_k$, then $$ N=(N\cap M_1)\cup\cdots\cup(N\cap M_k). $$ Viewing $N$ as a vector space in its own right, with $N\cap M_i$ subspaces of $N$, we see they cannot all be proper. So for some $i$, $N\cap M_i=N$, hence $N\subseteq M_i$.

In this problem, you can try pointing out that if $N$ is not a subset of $M_1$, then it must be a subset of $M_2$. I'll give you some hints on this:

• Since $N \not \subset M_1$, then there exists some $x \in N \ \backslash \ M_1$.

• You should notice that, since $N = M_1 \cup M_2$ (i.e, every element in $N$ must belong to either $M_1$, or $M_2$), and we have that, $x \in N \ \backslash \ M_1$, so $x \in M_2$.

• Now, can you try showing that for all $n \in N$, $n$ must belong to $M_2$? Hint: Take their sum, and consider 2 cases. ;)

To show it holds not only for $k = 2$, but for any $k \in \mathbb{N} \ \backslash \ \{ 0 \}$, you can try using Proof by Induction. Case $k = 1$ is trivial, and $k = 2$ has already been done, can you continue from here? :)