Ok, last problem today! I've already read any other posts on the sandwich principle, but I've never seen it before today, and my chapter doesn't explain it, just gives a question on it! (Wondering) If anyone could show me how this is done I would really appreciate it!
Let and be sequences with limits respectively, and suppose that, for all ,
Show that [This is sometimes called the sandwich principle. It can be useful if and are "known" sequences and is unknown. It is especially useful when , for in this case we conclude that .]
Nice explaination, if I did actually know what and were!In this case I don't know what to do, can anyone help please?
Letís do it for one half, you can expand.
Suppose then let .
Use the sequence convergence definition twice to get .
Removing the absolute values and using the given we get: .
You can see the contradiction.
Now do it again for the other half.
As you said, suppose that for all .
I give you an example. Usually the exercise will make you calcul the limit of the sequences and . Suppose now that you found that tends to 10 (from below) when tends to positive infinite and tends to 10 (from bottom) when tends to positive infinite. If you can see that and for a given sequence , then by the sandwich's theorem (or principle), tends to 10, which is very intuitive!