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., γ = -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 γ = 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.