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In this note we show that a $p$-complete nilpotent space $X$ has a$p$-complete suspension spectrum if and only if its homotopy groups $\pi_*X$are bounded $p$-torsion. In contrast, if $\pi_*X$ is not all bounded$p$-torsion, we locate uncountable rational vector spaces in the integralhomology and in the stable homotopy groups of $X$. To prove this, we establisha homological criterion for $p$-completeness of connective spectra. Moreover,we illustrate our results by studying the stable homotopy groups of$K(\mathbb{Z}_p,n)$ via Goodwillie calculus.

On the comparison of stable and unstable $p$-completion