Proof:

(i) Choose a Lebesgue number for the open cover $\{\sigma^{-1}(U)\;/ U \in {\cal U}\}$. According to theorem (34.3), the images of the simplicies occurring in the chain ${\cal B}^k\sigma$ are the same as the images under $\sigma$ of the affine simplicies occurring in $B^k\iota_p$, where $\iota_p$ is the identity map of $\Delta_p$. However, theorem (34.4) states that the simplicies occurring in $B^k\iota_p$ have diameters less than $(p(p+1)^{-1})^k$. Thus, if we choose $k$ sufficiently large the image of each of the simplicies in $B^k\sigma$ would lie in one of the open sets of ${\cal U}$.

To prove (ii) we use the naturality of ${\cal J}$ and proceed as in the proof of theorem (34.4). Let $\sigma:\Delta_p\longrightarrow X$ have its image in $U \in {\cal U}$. Then ${\cal J}\sigma = \sigma_{\sharp}(J\iota_p)$. But we see immediately from the definition of $J$ in theorem (34.3) that $J\iota_p$ is a $\mathbb{Z}-$linear combination:

$$J\iota_p = \sum c_k\lambda_k$$

where each $\lambda_k$ is a (degenerate) affine (p+1) simplex contained in $\Delta_p$ and hence $\sigma_{\sharp}(\lambda_k)$ is a singular $(p+1)$ simplex with image contained in $U$. $\square$