A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is routinely restricted to the non-Archimedean case.
For example, in the Archimedean fields $\mathbb{R}$ and $\mathbb{C}$ with absolute value, it follows from Cantor's Intersection Theorem that every decreasing sequence of closed balls also has non-empty intersection. This fails for $\mathbb{Q}$, since $[\pi-1/n,\pi+1/n]\cap\mathbb{Q}$ is a decreasing sequence of closed balls in $\mathbb{Q}$ with empty intersection. So it cannot be true for all Archimedean normed fields.
Is there some crucial distinction between the Archimedean and non-Archimedean case vis-à-vis the spherical completeness property?
Many thanks