= Solution
For weights summing to one, collect the exact <portfolio> return as
$$
r^{(n)}=a^{(n)}+\sum_{j=1}^m b_j^{(n)}f_j+e_n,\qquad
a^{(n)}=\sum_iw_i a_i,\quad b_j^{(n)}=\sum_iw_ib_{ij},\quad e_n=\sum_iw_i\epsilon_i.
$$
The desired approximation means that $e_n\to0$, for example in <mean-square convergence>. Centering the residuals gives $\mathbb Ee_n=0$, but its <variance> is
$$
\operatorname{Var}(e_n)=\sum_{i,k}w_iw_k\operatorname{Cov}(\epsilon_i,\epsilon_k).
$$
This identifies an omission in the printed hypotheses: bounded individual <variances> do not control the cross terms. Indeed, take $\epsilon_i=Z$ for every $i$, where $Z$ is a nondegenerate centered <random variable> with $\operatorname{Var}Z<s^2$. Equal weights $1/n<W/n$ for any $W>1$ retain $e_n=Z$ at every $n$. \b[The claimed diversification conclusion does not follow from the printed assumptions alone.]
Under the usual extra assumption that the residuals are pairwise uncorrelated and the weights are nonnegative, the advertised calculation is
$$
\mathbb Ee_n^2=\sum_iw_i^2\sigma_i^2
\le s^2\left(\max_iw_i\right)\sum_iw_i
\le\frac{s^2W}{n}\longrightarrow0.
$$
The <Chebyshev inequality> then gives $\Pr(|e_n|>\delta)\le s^2W/(n\delta^2)\to0$. Alternatively, signed weights satisfying $|w_i|\le W/n$ give the bound $s^2W^2/n$. These are instances of the <covariance criterion for diversification of factor residuals>. More generally, with that absolute weight bound, $\sum_{i,k}|\operatorname{Cov}(\epsilon_i,\epsilon_k)|=o(n^2)$ suffices.
If <short selling> are allowed, the printed one-sided weight bound is another insufficiency. Choose $0<c<W-1$, put $w_1=-c$ and $w_i=(1+c)/(n-1)$ for $i\ge2$, and let only $\epsilon_1$ be a nondegenerate centered residual. For all sufficiently large $n$ the one-sided bounds hold, but $e_n=-c\epsilon_1$ never vanishes. Thus nonnegative weights or an absolute bound is needed.
With the repaired assumptions, \b[well-diversified <portfolios> become approximately exposed only to the common factors], and their residual <standard deviation> is $O(n^{-1/2})$ in the uncorrelated case. The averaged coefficients may depend on $n$; convergence to fixed coefficients needs additional assumptions. In the associated asymptotic <arbitrage pricing theory>, diversifiable risk cannot command a persistent premium in well-diversified <portfolios> under absence of asymptotic <arbitrage>, while common-factor exposure can. This does not prove an exact pricing equation for every individual noisy asset from the finite-market hypotheses alone.
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