Properties of Sequences Here is a FITB proof arguing that a sequence cannot converge to two different numbers. The basic idea is to argue that if we assume this can happen, we deduce that something contradictory to something we already know. This is a valid form of argument, though perhaps strange when you see it first. It's not unlike Sherlock's 'when you have eliminated the impossible, whatever remains, however improbable, must be the truth'. The purpose here is to strenghten mathematical understanding and use of the definition of convergence. Theorem 0.1: (Uniqueness of Limits) If a sequence convergences, it must converge to a unique (just one!) value. Proof: Suppose a sequence S converges to two different numbers L and M, with L > M. Choose a number e to be e = L M, which is a particular e value with e > 0. Since S 4 converges to L, the number of terms of the sequence NOT in the interval (L e, L + e) is finite, and this means that there must be an infinite number of terms of the sequence inside the interval (L e, L + e). But none of THESE infinite number of terms are also inside the interval (M e, M + e), because M + e < L e (because of the way we chose e -- specificially e = L M. Easy to see this visually.) But this means that the 4 number of terms of the sequence not in the interval (M e, M + e) is INFINITE, which isn't supposed to happen because the sequence converges to M! Thus our original assumption must be wrong, namely that we cannot have a sequence converging to two different values. Using the same kinds of arguments we can prove the following facts about combining convergent sequences. We use the same notation as we did for limits of functions of real numbers, because the same properties work for limits and the idea is very similar. (However technically there is a difference -- here we are talking about limits of sequences which is really quite a different thing than the limit like. The basic theme here is that combining CONVERGENT sequence in various ways results in a convergent sequence that also converges, to the value you would expect. Note that this theorem says NOTHING about what happens when you combine sequences that are not convergent (DIVERGENT), and in general all kinds of crazy things can happen. Theorem 0.2: (Combining convergent sequences) If sequence ( a n ) converges to α and sequence ( ) converges to β, then 1. sequence ( a n + b n ) converges to α + β, and 2. sequence ( a n b n ) converges to α β, and 3. sequence ( a n b n ) converges to α β, and 4. if also, then seqence a β 0 ( n ) converges to α. b n β 5. For any real number C 0 (C ) converges to Cα. a n lim x 3 First ask yourself what the notation means -- how do you get each of the terms of the b n
sequence described by the notation ( + )? Normally we use limit notation when working with sequences, because the ideas of convergence of a sequence to a limit and the limiting value of a function as x a are similar and many properties are the same. For example the first property would be written: If lim n a n = α and lim n b n = β then lim n a n + b n = α + β. a n b n An intuitive way to think about the first property is as follows: Given two convergent sequences ( a n ) α and ( b n ) β. Since the sequences converge the terms a n get closer and closer (as close as we want!) to α, and the terms b n get closer and closer (as close as we want!) to β. Adding a number very close to α to a number very close to β will result in a number very close to α + β. Thus we expect the sequence ( a n + b n ) to converge to α + β, which it does!. Example: ( 1 0 and so the above theorem (second bullet) tells us that n ) (3 3 n=1 ) n=1 ( 1 3 0 3 n ) n=1 Suppose sequence ( a 1, a 2, a 3,... ) = ( a n ) converges to α. So intuitively the terms n=1 a n get closer and closer to α as n gets larger and larger. What happens if we drop the first few terms of the sequence? What does ( = (,,,... ) a n ) n=100 a 100 a 101 a 102 converge to, if anything? Clearly it still converges to α. In other words, what happens at the 'beginning' of a sequence doesn't matter -- it's the behaviour 'at the end' that matters. For this reason we sometimes do not write where a sequence starts: for example or (a n ). ( 1 3+n )