We show that non-damped acoustic plasmons exist in single wall carbon nanotubes (SWCNT) and propose that the non-damped acoustic plasmons may mediate electron-electron attraction and result in superconductivity in the SWCNT. The superconducting transition temperature Tc for the SWCNT (3,3) obtained by this mechanism agrees with the recent experimental result (Z. K. Tang et al, Science 292, 2462(2001)). We also show that it is possible to get higher Tc up to 99 K by doping the SWCNT (5,5).