Use Definition 11.3.1 to prove the following statements. (a) If the sequence an has a limit,...

 

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Use Definition 11.3.1 to prove the following statements. (a) If the sequence an has a limit, that limit is unique. (b) If an is a convergent sequence, then for every > 0, there exists an integer K such that for all m, n > K, we have an am| < . - Hint: in both cases, use the triangle inequality |x+yx+y DEFINITION 11.3.1 LIMIT OF A SEQUENCE lim an = 818 L if for each > 0, there exists a positive integer K such that if n K, then |an L| < . Use Definition 11.3.1 to prove the following statements. (a) If the sequence an has a limit, that limit is unique. (b) If an is a convergent sequence, then for every > 0, there exists an integer K such that for all m, n > K, we have an am| < . - Hint: in both cases, use the triangle inequality |x+yx+y DEFINITION 11.3.1 LIMIT OF A SEQUENCE lim an = 818 L if for each > 0, there exists a positive integer K such that if n K, then |an L| < .