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notes2 - CM221A ANALYSIS I NOTES ON WEEK 2 You must...

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CM221A ANALYSIS I NOTES ON WEEK 2 You must remember and be able to use all the definitions and theorems stated in this week’s notes. Their proofs can be found in the CM115 lecture notes. The proofs are not examinable. CONVERGENT SEQUENCES Let { a n } be a sequence. The Crucial Definition . We say that { a n } converges to a number a and write lim n →∞ a n = a or a n a if ε > 0 n ε N : n n ε ⇒ | a n - a | < ε. In usual words, this means the following: for every positive ε there is a positive integer n ε such that | a n - a | < ε whenever n n ε . The number a is called the limit of { a n } . A sequence which has a limit is called a convergent sequence. One can reformulate the above definition in “geometric” terms. The Crucial Definition: another version . The sequence { a n } converges to a number a if for every ε > 0 there are only finitely many elements a n lying outside the open interval ( a - ε, a + ε ). Remark. There are divergent sequences, that is, the sequences which do not converge to a limit. For instance, the sequence { +1 , - 1 , +1 , - 1 , +1 , . . . } does not converge. Definition. We say that { a n } converges to + and write a n + if R > 0 n R N : n n R a n > R.
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