Prove: If lim_(n => infty)a_n=L and a_n>a for all n then L >= a

Freddy Friedman 2022-07-17 Answered
Prove: If lim n a n = L and a n > a for all n then L a
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Answers (1)

eishale2n
Answered 2022-07-18 Author has 15 answers
Suppose that L < a. We put ε = a L > 0. Since lim n a n = L , there exists N 0 N such that
| a n L | < ε n N 0 .
Then a n L < a L for all n N 0 , or a n < L. This is contradict to the assumption a n > a for all n. Hence L a. In the above argument, we have seen that we only need a n > L for sufficiently large n.
Moreover, in the general case we do not have L > a. Indeed, we observe that although a n = 1 n > 0 for all n but L = lim n a n = lim n 1 n = 0
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