Let {a_n}, {b_n} and {c_n} be a 3 closed sequences. Prove the generalization of sandwich rule:

If a_n <= b_n <= c_n for all natural n and if liminf(a_n)=limsup(c_n)=L, then all the three sequences areconvergentto L.

I need the exact proof.

Thank you very much!