= Solution
For the <unit-width box kernel>, the order-zero <local polynomial estimator> minimizes, over constants $a$,
$$
Q_x(a)=\sum_{i=1}^nK\left(\frac{x_i-x}{h}\right)(Y_i-a)^2.
$$
A common factor such as $h^{-1}$ in these weights does not change the minimizer. Put $I_x=\{i:|i/n-x|\le h/2\}$ and $N_x=|I_x|$. If $N_x>0$, differentiating the quadratic gives its unique minimizer:
$$
\boxed{\widehat m_{n,h}^P(x)=\frac1{N_x}\sum_{i\in I_x}Y_i=\frac{\sum_iK((x_i-x)/h)Y_i}{\sum_iK((x_i-x)/h)}=\widehat m_{n,h}^K(x).}
$$
The last expression is the <Nadaraya–Watson estimator>. Every $x\in[0,1]$ is within $1/n$ of some design point $i/n$. Since $nh\ge2$ gives $h/2\ge1/n$, that point is in the closed kernel window. Thus $N_x\ge1$ and the equality holds throughout the domain, including $x=0$ and $x=1$.
For the error bound, first take the usual bandwidth range $0<h\le1$. The clipped window $[x-h/2,x+h/2]\cap[0,1]$ has length at least $h/2$. Any closed interval of length $L$ in $[0,1]$ contains at least $\lfloor nL\rfloor$ points of the design $\{1/n,\ldots,1\}$. Consequently
$$
N_x\ge\lfloor nh/2\rfloor\ge nh/4,
$$
where the last inequality uses $nh/2\ge1$. This is a <window occupancy for an equally spaced regression design> bound that remains valid at the boundary.
Let $L_m=\|m'\|_\infty$, the <Lipschitz constant> supplied by the bounded-derivative assumption. The <bias> is bounded by
$$
|\mathbb E\widehat m_{n,h}^P(x)-m(x)|\le\frac1{N_x}\sum_{i\in I_x}|m(x_i)-m(x)|\le L_mh/2.
$$
Independence and the variance bound give
$$
\operatorname{Var}(\widehat m_{n,h}^P(x))=\frac1{N_x^2}\sum_{i\in I_x}V(x_i)\le\frac{\sigma^2}{N_x}.
$$
By the <Cauchy-Schwarz inequality> and the triangle inequality,
$$
\boxed{\mathbb E|\widehat m_{n,h}^P(x)-m(x)|\le\frac{2\sigma}{\sqrt{nh}}+\frac12\|m'\|_\infty h\le2\left(\frac{\sigma}{\sqrt{nh}}+\|m'\|_\infty h\right).}
$$
Thus \b[$\kappa=2$ works uniformly in $x$ and the design size in this bandwidth range]. The same clipped-length proof in fact works for $0<h\le2$.
There is a genuine omission in the unrestricted formulation: an upper restriction on bandwidth is necessary for the displayed $1/\sqrt{nh}$ variance scale. Take $n=1$, $m\equiv0$, $Y_1\sim N(0,\sigma^2)$ and $h\ge2$. The window contains the sole observation for every $x\in[0,1]$, and $nh\ge2$ holds. Its mean absolute error is $\sigma\sqrt{2/\pi}$, whereas the proposed variance term is $\kappa\sigma/\sqrt h$. No fixed $\kappa$ can satisfy this for unbounded $h$.
A valid statement for every $h>0$ satisfying $nh\ge2$ uses $q=\min(h,1)$. The clipped window has length at least $q/2$, so $N_x\ge nq/2-1$, while the preceding coverage argument gives $N_x\ge1$. If $nq\ge4$, the first bound gives $N_x\ge nq/4$; if $nq<4$, the second does. Therefore the same bias and variance calculation proves the unrestricted correction
$$
\boxed{\mathbb E|\widehat m_{n,h}^P(x)-m(x)|\le\frac{2\sigma}{\sqrt{n\min(h,1)}}+\frac12\|m'\|_\infty h.}
$$
This <mean absolute error of local constant regression> bound reduces to the requested rate for ordinary small bandwidths and saturates its stochastic term when the window covers the whole design.
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