Thread: Sandwich theorem generalization

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 are convergent to L.


I need the exact proof.
Thank you very much!

Well



hence , so is convergent and .

By a very similar argument you get that converges to . Now just apply the usual sandwich theorem. Hope this helps!