Regular　convergence,　together　with　other　types　of　convergence,　have　been　studied　since　the　1970s　for　discrete　approximations　of　linear　operators.　In　this　paper,　we　consider　the　eigenvalue　approximation　of　a　compact　operator　T　that　can　be　written　as　an　eigenvalue　problem　of　a　holomorphic　Fredholm　operator　function　F(&　eta;)　=　T-　&　eta;　1　I.　Focusing　on　finite　element　methods　(conforming,　discontinuous　Galerkin,　non-conforming,　etc.),　we　show　that　the　regular　convergence　of　the　discrete　holomorphic　operator　functions　F,,,　to　F　follows　from　the　compact　convergence　of　the　discrete　operators　T,,,　to　T.　The　convergence　of　the　eigenvalues　is　then　obtained　using　abstract　approximation　theory　for　the　eigenvalue　problems　of　holomorphic　Fredholm　operator　functions.　The　result　can　be　used　to　prove　the　convergence　of　various　finite　element　methods　for　eigenvalue　problems　such　as　the　Dirichlet　eigenvalue　problem　and　the　biharmonic　eigenvalue　problem.