Questions on cauchy definition of limit

• Mar 30th 2013, 01:55 AM
rokman54
Questions on cauchy definition of limit
I don't understand this proof. Can you clarify this for me?

Suppose {an}converges, then the limit is unique.

pf) Suppose {an} converges to $\alpha, \beta$ at the same time.

Take any epsilon > 0,

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

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

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

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

Therefore, alpha = beta
• Mar 30th 2013, 02:37 AM
Gusbob
Re: Questions on cauchy definition of limit
If you take the smaller of the two $N_i$, one of your inequalities $|a_n-\alpha|\leq \frac{\epsilon}{2}$ or $|a_n-\beta|\leq \frac{\epsilon}{2}$ might not hold, since this is only guaranteed for $n\geq N_1$ and $n\geq N_2$.
• Mar 30th 2013, 06:11 AM
HallsofIvy
Re: Questions on cauchy definition of limit
The definition of convergence involves "if n> N". By taking N equal to the larger of $N_1$ and $N_2$, " $n> N$" gives both " $n> N_1$" and " $n> N_2$".