Limit Theorems in Real Analysis: Characterization of Limit using Open Intervals, Study notes of Mathematics

Download Limit Theorems in Real Analysis: Characterization of Limit using Open Intervals and more Study notes Mathematics in PDF only on Docsity! Math 4200 Theorem: Let { } n a be a sequence of real numbers. Then lim n n a a →∞ = if and only if given any open interval ( , )r s containing a, ( , )r s contains all but finitely many terms of the sequence { } n a . Proof: I. Suppose lim n n a a →∞ = . Let ( , )r s be an open interval containing a. Let min{ , }a r s aε = − − . There exists 0M > such that if n M> then n a a ε− < . If n M> , then n a a aε ε− < < + . If n M> , then ( ) ( ) n a a r a a s a− − < < + − . If n M> , then n r a s< < . Therefore, ( , )r s contains all but finitely many terms of the sequence. II. Suppose that given any open interval ( , )r s containing a, ( , )r s contains all but finitely many terms of the sequence { } n a . Let 0ε > . Consider the interval ( , )a aε ε− + . If all but finitely many terms of the sequence are contained in this interval, then there exists 0M > such that if n M> then n a a aε ε− < < + . So, there exists 0M > such that if n M> then n a a ε− < .