Question: Show that Zp is isomorphic to the p-adic completion of Z; that is, the completion of Z when Z is considered a metric space via the p-adic metric.
I'm stuck. If we take an element a in Zp, i.e. a = a1, a2, a3, ... Considering a sequence of integers b1, b2, b3, ... such that bi \equiv ai (mod p^i), how do we show that such a sequence is Cauchy in Z?
Further, how do we show that for a given a, all such sequences are equivalent?
You can show that your sequence is Cauchy without too much trouble. If $i>j \ge N$, then since $b_i \equiv a_i \mod p^i$, then $b_i \equiv a_i \equiv a_j \equiv b_j \mod p^j$. Therefore $p^j$ and $p^N$ both divide $b_i-b_j$.
If you take two different sequences of integers $\{b_i\}, \{c_i\}$ representing the same element of the inverse system, then $c_i-b_i \equiv 0\mod p^i$ for all $i$, and therefore this difference is clearly a sequence converging to zero.