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 $\displaystyle \alpha, \beta$ at the same time.

Take any epsilon > 0,

there exists N1 $\displaystyle \in $ N such that $\displaystyle \mid a_n- \alpha \mid \leq $ epsilon/2

there exists N2 $\displaystyle \in $ N such that $\displaystyle \mid a_n- \beta \mid \leq $ epsilon/2

Suppose N = Max(N1 , N2). <---- I don't understand this part. Why do you take the maximum of N1 and N2?

$\displaystyle \mid \alpha - \beta \mid = \mid \alpha - a_n+ a_n - \beta \mid \leq \mid \alpha - a_n\mid + \mid a_n- \beta \mid \leq \epsilon $

Therefore, alpha = beta

Re: Questions on cauchy definition of limit

If you take the smaller of the two $\displaystyle N_i$, one of your inequalities $\displaystyle |a_n-\alpha|\leq \frac{\epsilon}{2}$ or $\displaystyle |a_n-\beta|\leq \frac{\epsilon}{2}$ might not hold, since this is only guaranteed for $\displaystyle n\geq N_1$ and $\displaystyle n\geq N_2$.

Re: Questions on cauchy definition of limit

The definition of convergence involves "if n> N". By taking N equal to the **larger** of $\displaystyle N_1$ and $\displaystyle N_2$, "$\displaystyle n> N$" gives both "$\displaystyle n> N_1$" and "$\displaystyle n> N_2$".