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={−n1∣∣n∈N}, then supA=0, but a<0 for every a∈A
2)True
n→∞liman=L⇔∀ε>0∃N∈N∀n>N∣an−L∣<ε
n→∞lim∣an−L∣=0⇔∀ε>0∃N∈N∀n>N∣∣an−L∣−0∣<ε
Rightsides of the first and the second statements are equivalent, so leftsides of the statements are equivalent.
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