Sequence Limits

1. In the following, let (or ) be a sequence of real numbers and L is a real number. Rewrite each statement symbolically using quantifiers.
   1. Definition: The sequence s is said to converge to L provided that if then there is a number N such that n>N implies that . In this case, s is called a convergent sequence, L is its limit, and we write .
   2. Definition: The sequence s is said to converge if it converges to some number L.
2. For each of the following sequences, prove whether or not it converges.
   1. .
   2. .
   3. .
   4. .
   5. .
   6. .
   7. .
   8. .
   9. where [x] denotes the greatest integer not exceeding x.
   10. , where [x] is as in the previous problem.
   11. .
   12. if n is odd and if n is even.
   13. if n is odd and if n is odd.
   14. if n is odd and if n is even.
3. Prove.
   1. If s is a convergent sequence, then s is bounded.
   2. If s is a convergent sequence, then its limit is unique.
   3. If s and t are sequences such that and for every n, , then .
   4. If and for every n, is in the interval [a,b], then L is also in [a,b].
   5. If s is a sequence and L is a number, then if and only if .
   6. If s is a bounded sequence and t is a null sequence (a sequence converging to 0), then st is a null sequence.
   7. If t is a convergent sequence whose limit is not zero, and for n in , , then 1/t is a bounded sequence.
   8. Suppose each of s and t is a convergent sequence and c is a number; then s+t, s-t, and cs are convergent. Also, if and , then and .
   9. Suppose each of s and t is a convergent sequence, say and ; then st is convergent and . If, in addition, is never 0 and , then s/t is convergent and .