\textit{I. B. Frenkel}, \textit{H. Garland} and \textit{G. J. Zuckerman} (FGZ) [Proc. Natl. Acad. Sci. USA 83, 8442-8446 (1986; Zbl 0607.17007)] computed the BRST cohomology of the open bosonic string after identifying it with a particular semi-infinite cohomology for which they had proven a vanishing theorem that the BRST cohomology is zero except at zero ghost number. This vanishing theorem is applicable not only to the particular representation of the Virasoro algebra but to a large class of graded Lie algebras and their representations. In the present paper the FGZ result is extended to the representations of the \(N=1\) super-Virasoro algebras (i.e., the Neveu-Schwarz and Ramond superalgebras) and then to the Neveu- Schwarz-Ramond (NSR) string [\textit{M. B. Green}, \textit{J. H. Schwarz} and \textit{E. Witten}, Superstring Theory (Cambridge Univ. Press, 1987; Zbl 0619.53002)]. The authors prove vanishing theorems for the relative BRST subcomplexes for the Neveu-Schwarz and Ramond sectors and then for the full BRST complex. The proof uses the algebraic machinery of spectral sequences of differential complexes [\textit{S. Lang}, Algebra (Addison-Wesley, 1984; Zbl 0712.00001); \textit{P. Griffiths} and \textit{J. Harris}, Principles of Algebraic Geometry (New York, Wiley, 1978; Zbl 0408.14001)]. Here a differential complex consists of a pair (E,d) of a vector space E and a linear map d: \(E\to E\), called the differential, obeying \(d^ 2=0\), and a spectral sequence is a sequence \((E_ r,d_ r)_{r=0,1,...}\) of differential complexes, where \(E_{r+1}\) is the cohomology space of the preceding complex \((E_ r,d_ r)\), i.e., \(E_{r+1}=\ker d_ r/im d_ r\). Using these results the authors give simple proofs of the ``no- ghost'' theorems for both sectors of the NSR string.