Let A be a \(C^*\)-algebra (non-unital, in general). A net \((f_{\alpha})\) of pure states approaches infinity if lim \(f_{\alpha}(a)=0\) for every a in A; a net \((p_{\alpha})\) of projections in \({\mathcal U}(A)\), the universally measurable elements in \(A^{**}\) [see G. K. Pedersen’s \(``C^*\)-algebras and their automorphisms groups” (1979; Zbl 0416.46043)] approaches infinity if lim \(\| ap_{\alpha}\| =0\) for every a in A. This paper studies the relationship between these two notions. The main result of the paper is the following. Let A be separable, \((p_ n)\) a sequence of pairwise orthogonal, minimal projections in \(A^{**}\) supporting the pure states \((f_ n)\). Then \((f_ n)\) approaches infinity if and only if \((p_ n)\) approaches infinity. As an application it is shown that, given a sequence \(\{f_ n\}\) of pure states in A, a maximal abelian \(C^*\)-subalgebra C of A can be found such that \(f_ n| C\) is a pure state on C with unique state extension to A.