For the purpose this question, let us only consider those topological spaces which are paracompact and Hausdorff. For a space $X$, we define the Cech cohomology as $\check{H}(X,\mathbb Z):=\displaystyle\lim_{\to} H^\bullet(\mathcal V,\mathbb Z)$ where $\mathcal V$ runs over all open covers of $X$ (ordered by refinement) and $H^\bullet(\mathcal V,\mathbb Z)$ denotes the simplicial cohomology of the nerve of $\mathcal V$. It is well-known that this coincides with the (derived functor) cohomology $H^\bullet(X,\underline{\mathbb Z}_X)$, where $\underline{\mathbb Z}_X$ is the constant sheaf with stalks $\mathbb Z$ on $X$.
In his book "Lectures on Algebraic Topology", Dold defines Cech cohomology (only for locally compact subsets of Euclidean Neighborhood Retracts) $X$ as follows. First choose an embedding $\iota: X\subset E$ into an ENR $E$, and then define $\check{H}(X,\mathbb Z):=\displaystyle\lim_{\to} H^\bullet(U,\mathbb Z)$ where $U$ ranges over the collection of open neighborhoods of $\iota(X)$ in $E$ (ordered by reverse inclusion), and $H^\bullet(U,\mathbb Z)$ is the singular cohomology of $U$.
How can I see that the two definitions are the same? I am ok with a proof even in the special case when $E$ is a topological manifold, as this is the case I am interested in at the moment. I'm also willing to accept the fact that for locally contractible spaces, the Cech cohomology (first definition) is naturally isomorphic with singular cohomology.