Approximating step functions by Haar wavelets

Question

Let $\psi = \chi_{[0,1/2)} - \chi_{[1/2,1)}$, then $\psi_{n,k}(t) = 2^{n/2}\psi(2^nt-k)$ with $n \in \mathbb{N}$ and $k \in \{0,1,\dots,2^n-1\}$ defines the Haar-Wavelets on $L^2(0,1)$. Let $S$ be the set of all those Wavelets. I already showed that $S$ is an orthonormal system, which is quite straightforward. To show that they form an orthonormal basis, my idea is the following:
The step functions (functions of the form $\sum_i^N a_i \chi_{Q_i}$ a.e. for $Q_i$ half-open intervals) are dense in $L^2(0,1)$. If I can approximate any function $\chi_{Q_i}$ by a linear combination of Haar-Wavelets I am done, because then we have $\overline{\text{lin } S} = L^2(0,1)$. Since $\text{supp }\psi_{n,k}$ gets arbitrarily small I can find $N \in \mathbb{N}$ and an index set $I$ such that $\lambda(Q_i \setminus \bigcup_{j \in I} \text{supp }\psi_{N,j}) < \varepsilon$. But I can't find a way to get rid of the parts of this linear combination where it has the value $-1$ in order to get my desired approximation.
Can someone give me a hint, please?