P-adic numbers complete/incomplete P-adic numbers are complete in one sense and incomplete in another sense. Is it so?
Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for p-adics because they are not connected. (if I am correct).
It seems I need elaboration of this "It can be shown that the rationals, together with the $p$-adic metric, do not form a complete metric space. The completion of this space can therefore be constructed, and the set of $p$-adic numbers  $\mathbb{Q}_p$is defined to be this completed space."Math World.
How is the completion constructed with same metric and same numbers?

Alternatively, one can verify that, e.g., the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $$x_n = \sum_{j = 0}^n{p^{j^2}}$$ is Cauchy with respect to the $p$-adic valuation, but does not have a limit in $\mathbb{Q}$.
