Answer to Question #152504 in Real Analysis for Cypress

Question

Is the following statements True or False? Please state your answer and explain them.
1) If a < L for every element in a set A, then sup A < L.
2) limit of (an) when n goest to infinity is L if and only if the limit of |an - L| when n goes to infinity is 0.

Answer 1

1)False

Let $A=\left\{-\frac{1}{n}\bigl | n\in\mathbb{N}\right\}$, then $\sup A=0$, but $a<0$ for every $a\in A$

2)True

$\lim\limits_{n\to\infty}a_n=L\Leftrightarrow\forall\varepsilon>0 \ \exists N\in\mathbb{N} \ \forall n>N \ |a_n-L|<\varepsilon$

$\lim\limits_{n\to\infty}|a_n-L|=0\Leftrightarrow\forall\varepsilon>0 \ \exists N\in\mathbb{N} \ \forall n>N \ \left||a_n-L|-0\right|<\varepsilon$

Rightsides of the first and the second statements are equivalent, so leftsides of the statements are equivalent.