layer as $wx$. With a limited number of distinct weight values, we can construct the weight we need by cascade and combination.

For clarity, we first consider $w\geq0$ and $x\geq0$, and relax these assumptions later. The connections with $w=0$ can be seen as an empty sub-network while $w=1$ can be easily implemented by 4 units with weight $\frac{1}{2}$. Now we show how to represent all integral multiples of $2^{-t}$ from $2^{-t}$ to $ 1-2^{-t}$, which will lead to the statement (1) by choosing the nearest one from $w$ as $w'$. Without loss of generality, we assume $t^{\frac{1}{\lambda-1}}$ is an integer. We use $\lambda$ weights that include $-\frac{1}{2}$ and $W$: $$W\triangleq\{2^{-1},2^{-t^{\frac{1}{\lambda-1}}},2^{-t^{\frac{2}{\lambda-1}}},\cdots,2^{-t^{\frac{\lambda-2}{\lambda-1}}}\}.$$ We first construct all $w$ from $W_c$ which is defined as $$W_c\triangleq\{2^{-1},2^{-2},\cdots,2^{-(t-1)}\}.$$ Similar to a numeral system with radix equal to $t^{\frac{1}{\lambda-1}}$, any $w_i\in W_c$ can be obtained by concatenating weights from $W$ while every weights in $W$ is used no greater than $t^{\frac{1}{\lambda-1}}-1$ times.

After that, all integral multiples of $2^{-t}$ from $2^{-t}$ to $ 1-2^{-t}$ can be represented by a binary expansion on $W_c$. Note that connections in the last layer for binary expansion use weight $\frac{1}{2}$, thus additional $2^{-1}$ is multiplied to scale the resolution from $2^{-(t-1)}$ 