Chapter 10.1, Problem 74E

To determine

That the limits of sequence are unique.

Answer to Problem 74E

The only nonnegative number smaller than every positive number is 0, so $\left|L_2-L_1\right|=0$ or $L_1=L_2$ .

Explanation of Solution

Given information:

The sequence {a_n} converges to the number L if to every positive number $\epsilon$ there corresponds an integer N such that for all n,

  $n>N\Rightarrow \left|a_n-L\right|<\epsilon$

Formula used:

  $\left|L_2-L_1\right|=\left|L_2-a_n+a_n-L_1\right|$

Calculation:

Given an $\in >0$ , by definition of convergence, for $\in$ there corresponds N such that for all

  $n>N\Rightarrow \left|L_1-a_n\right|<\in \left|L_2-a_n\right|<\in$

Now,

  $\begin{array}{l}\left|L_2-L_1\right|=\left|L_2-a_n+a_n-L_1\right|\\ \left|L_2-L_1\right|\le \left|L_2-a_n\right|+\left|a_n-L_1\right|<\in +\in =2\in \end{array}$

  $\left|L_2-L_1\right|<2\in$ , says that the difference between the two fixed values is smaller than any positive number 2 $\in$

The only nonnegative number smaller than every positive number is 0, so $\left|L_2-L_1\right|=0$ or $L_1=L_2$

Conclusion:

The only nonnegative number smaller than every positive number is 0, so $\left|L_2-L_1\right|=0$ or $L_1=L_2$ .