I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields".
For an algebraic number field $K$ let $E$ denote the global units, for each prime $\mathfrak p$ over $p$ let $U_\mathfrak p$ be the local units of the completion $K_\mathfrak p$, and let $U_{1,\mathfrak p}$ be the principal units, i.e. the units $\varepsilon \equiv 1$ modulo $\mathfrak p$. Let $U = \prod_{\mathfrak p|p} U_\mathfrak p \quad \text{and} \quad U_1 = \prod_{\mathfrak p|p} U_{1,\mathfrak p}.$ The units $E$ are embedded diagonally in $U$ and the units $\varepsilon$ that embed to $U_1$ are denoted by $E_1$. Then $E_1$ is a $\mathbb Z$-module of rank $r_1+r_2-1$ (where $r_1$ is the number of real embeddings and $2r_2$ is the number of complex embeddings). Leopoldt's conjecture says that $\overline{E_1}$ (the closure of $E_1$ in the topology of $U_1$) is a $\mathbb Z_p$-module of rank $r_1+r_2-1$
Now in the proof of Washington's theorem 5.31 (and the example before that) he uses the fact that a $\mathbb Z$-basis of $E_1$ modulo torsion generates $\overline{E_1}$ modulo torsion as a $\mathbb Z_p$-module. This seems plausible to me, however I haven't been able to prove it.
So my question is: Why does a $\mathbb Z$-basis of $E_1$ (modulo torsion) generate $\overline{E_1}$ over $\mathbb Z_p$ (modulo torsion)?