How the surface state (SS) develops and how the spin transport in a curved cylindrical topological insulator nanowire have attracted theoretical attention recently. However, experimental confirmation for the SS in such a real modeling system still remains insufficient. Here we carried out a systematic comparative study on the cylindrical single-crystal Bi2Te3 nanowires of various diameters, and report unambiguously dual evidence for the Dirac SS. Both the predicted anomalous Aharonov-Bohm (AB) quantum oscillations with a period of h/e in H// and the 1/2-shifted Shubnikov-de Haas (SdH) oscillations (i.e., = ？1/2) in H⊥ were indentified below 1.4？K. In addition, Altshuler-Aronov-Spivak (AAS)-like oscillations with a period of h/2e and ordinary SdH oscillations with = 0 were also resolved. These data provide clear evidence of coexistence of the nontrivial topological Dirac state and trivial electron state on the surface of topological insulator nanowire.