Proof verification: Every null set is a subset of an uncountable null set

Question

Let $\lambda$ be Lebesgue measure on $\mathbb{R}^n$.

Is it true that every null set, i.e $\lambda(N) = 0$ is a subset of an uncountable null set?

I think so, because every $k$-dimensional subspace $S$ of $\mathbb{R}^n$ has measure zero and if $\{v_1,...,v_k\}$ is its basis then $S = \text{span}(v_1,...,v_k)$ implying that $S$ is uncountable, since the scalar field is $R$ uncountable. For any null set $N \subset N \cup S$. If $n = 1$ we take $N \subset N \cup C$ where $C$ is Cantor-set. Is that correct?

Answer 1

If $n=1,$ and $K$ is the Cantor set, then $K \cup N$ fits the bill. If $n>1,$ then $\{0\}\times \mathbb R^{n-1}\cup N$ does the job.