Question #152504
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.
1
Expert's answer
2020-12-24T12:46:52-0500

1)False

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

2)True

limnan=Lε>0 NN n>N anL<ε\lim\limits_{n\to\infty}a_n=L\Leftrightarrow\forall\varepsilon>0 \ \exists N\in\mathbb{N} \ \forall n>N \ |a_n-L|<\varepsilon

limnanL=0ε>0 NN n>N anL0<ε\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.


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