If you take the smaller of the two , one of your inequalities or might not hold, since this is only guaranteed for and .
I don't understand this proof. Can you clarify this for me?
Suppose {a_{n}}converges, then the limit is unique.
pf) Suppose {a_{n}} converges to at the same time.
Take any epsilon > 0,
there exists N1 N such that epsilon/2
there exists N2 N such that epsilon/2
Suppose N = Max(N1 , N2). <---- I don't understand this part. Why do you take the maximum of N1 and N2?
Therefore, alpha = beta