Originally Posted by

**Melsi** Hello,

I would like to ask if the proof below, concerning the squeeze theorem is correct. I guess someone else must have come up with it but I didn't find anything in inte rnet...

Gn =< An =< Bn ... where sequences (Gn), (Bn) -> a, then a is a limit for (An) as well.

[=< less or equal, -> converges to, <=> equivalent]

Proof

-Consider sequences (Xn), (Yn) of positive or zero terms! (1)

Gn=<An <=> Gn=An-Xn <=> |Gn-a|=|An-Xn-a| <=> |An-Xn-a|<e <=> |An-(a+Xn)|<e <=> An->(a+Xn)

This makes no sense: in order to obtain the limit of $\displaystyle A_n$ you must make $\displaystyle n\to\infty$ , so

how

come n __still__ appears in the expression $\displaystyle A_n\xrightarrow [n\to\infty]{} a+X_n$ ??!

The same can be said about the following step, too.

Tonio

Bn>=An <=> Bn=An+Yn <=> |Bn-a|=|An+Yn-a| <=> |An+Yn-a|<e <=>

|An -(a-Yn)|<e <=> An->(a-Yn)

This is:

{An->(a+Xn) , An->(a-Yn)} => a+Xn=a-Yn <=> Xn=Yn=0 because follows from (1) they are positive or zero.

Please, if anyone finds that there is a mistake, or it is correct, I would be very glad to know.

Thank you all, in advance!