Let $\mathbb{Z}_p= {a\in\mathbb{Q}_p : a=\sum_{i=0}^{\infty}d_ip^{i},0\leq d_i\leq p-1\;\forall\;i\geq 1}$ be a subset of $\mathbb{Q}_p.$ In other words, $\mathbb{Z}_p$ is the set of all p-adic numbers whose canonical p-adic expansion starts at a non-negative power of p. This set is called the p-adic integers. Prove that $\mathbb{Z}_p = {a\in\mathbb{Q}_p:|a|_p\leq 1}$.
Please help me with this proof. I have no idea where to start. Thanks.
You have defined $\Bbb Z_p$ to be a subset of $\Bbb Q_p$, so all that’s needed is a proof that $z\in\Bbb Q_p$ with $|z|\le1$ implies that $z\in\Bbb Z_p$. You seem to have defined $\Bbb Q_p$ via canonical expansions, and it should be clear (since $m<0$ implies that $|p^m|>1$) that if $|z|\le1$, then there are only nonnegative powers of $p$ in its canonical expansion.