the sub-network that approximates any weight in Section \[weights\], and finally the approximation of general functions and our main results are in Section \[general\].

Approximation of squaring/multiplication {#sqaure}
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\[p1\] Denote the design parameter that determines the approximation error bound as $r$. Let $f_s^r$ be a ReLU sub-network with only two weight values $\frac{1}{2}$ and $-\frac{1}{2}$. The function $f_s(x)=x^2$ on the segment $[0,1]$ can be approximated by $f_s^r$, such that (1) if $x=0$, $f_s^r(x) = 0$; (2) the approximation error $\epsilon_s\leq 2^{-2(r+1)}$; (3) the depth is $\mathcal{O}\left(r\right)$; (4) the width is a constant; (5) the number of weight is $\mathcal{O}\left(r\right)$.

The proof and the details of the sub-network constructed are included in Appendix \[PP1\]. Once the approximation to squaring function is obtained, we get Proposition \[p2\] by the fact that $2xy=(x+y)^2-x^2-y^2$.

\[p2\] Denote the design parameter that determines the approximation error bound as $r$. Given $x\in[-1,1]$, $y\in[-1,1]$, and only two weight values $\frac{1}{2}$ and $-\frac{1}{2}$, there is a ReLU sub-network with two input units that implements a function $\times'$: $\mathbb{R}^2 \mapsto \mathbb{R}$, such that (1) if $x=0$ or $y=0$, then $\times'(x,y)=0$; (2) for any $x$, $y$, the error $\epsilon_{\times'}=|\times'(x,y)-xy| \leq 6\cdot2^{-2(r+1)}$; (3) the depth is $\mathcal{O}\left(r\right)$; (4) the width is 