Math 104, Fall 07
Homework#:6 Sequences
1. Give an example of a metric space which is not complete, i.e., in which there exists a
Cauchy sequence which is not converge.
2. Let (X:d)be a compact metric space. Show that Xis complete.
3. Let Xbe a complete metric space. Show that every closed subset FXis also
complete, i.e., show that if (xn)Fis a Cauchy sequence then it has a limit Lin F:
4. Consider the sequence xn+1 =p2 + xnwith x1= 1:Show that this sequence converge
and compute its limit.
5. Show that a convergent sequence in a metric space is bounded.
Good luck!!
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