MAT 472 — Spielberg HOMEWORK 4 Due: 9/21/06
Write neatly, not too small, and not too lightly. You may discuss the problems with other
students from class, but you must write your own solutions. Reread your proofs before
copying them out to turn in; I really do mean that you should write (at least) one draft of
each solution.
13. Let (an)∞
n=1 and (bn)∞
n=1 be sequences of real numbers that are bounded. Prove that
lim sup
n→∞
(an+bn)≤lim sup
n→∞
an+ lim sup
n→∞
bn,
and that equality holds if (an) or (bn) is convergent.
14. Let (an)∞
n=1 be a bounded sequence of real numbers. Let
A={x∈R:an< x for infinitely many n}.
Prove that inf A= lim inf n→∞ an.
15. Let (X, d) be a metric space. For a nonempty subset E⊆Xdefine the diameter of E
by
diam (E) = sup {d(x, y) : x, y ∈E}
(where the diameter of an unbounded set is defined to be +∞).
(i) Prove that diam (E)<∞if and only if diam (E)<∞, for any nonempty E⊆X.
(ii) Prove that diam (E) = diam (E) for any nonempty E⊆X.
16. Let (X, d) be a metric space. Prove that Xis complete if and only if the following
condition holds: for every decreasing sequence F1⊇F2⊇ · · · of nonempty closed subsets
of Xwith limn→∞ diam (Fn) = 0, there exists an element a∈Xsuch that
∞
\
n=1
Fn={a}.