Questions on cauchy definition of limit

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

Re: Questions on cauchy definition of limit

If you take the smaller of the two , one of your inequalities or might not hold, since this is only guaranteed for and .

Re: Questions on cauchy definition of limit

The definition of convergence involves "if n> N". By taking N equal to the **larger** of and , " " gives both " " and " ".