# Sandwich theorem generalization

• December 4th 2009, 03:05 PM
Also sprach Zarathustra
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!
• December 4th 2009, 03:09 PM
Bruno J.
Well

$L=\liminf a_n \leq \limsup a_n \leq \limsup c_n = L$

hence $\limsup a_n = \liminf a_n = L$, so $\{a_n\}$ is convergent and $\lim a_n = L$.

By a very similar argument you get that $\{c_n\}$ converges to $L$. Now just apply the usual sandwich theorem. Hope this helps!