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Proof: Upper and lower limits

Please see the attached file for the fully formatted problems.

Let be a sequence of real numbers.

We define


I'm having trouble with the following three proofs:

1) Show that
2) Show that if the limit of only exists when , then .
3) Show that if , then the limit exists, and .


Solution Summary

Proofs are provided for three statements involving limits. The solution is detailed and well presented.