MAT 472 HOMEWORK 4 — REVISED Due: 9/23/04 (Thursday)
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. If you submit some nonsense in the course of a proof, the whole thing
might strike me as not worth reading.
16. Let (an)∞
n=1 be a bounded sequence of real numbers.
(i) Prove that
lim inf
n→∞
an≤lim sup
n→∞
an.
(ii) Prove that equality holds in (i) if and only if (an) is convergent, and that in this case
(an) converges to the common value in (i).
17. 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.
18. 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 infn→∞ an.
19. (i) Let (xn) and (yn) be Cauchy sequences in a metric space (X, d). Prove that the
sequence
d(xn, yn)∞
n=1
converges in R.
(ii) If (xn)→xand (yn)→yprove that the sequence in (i) converges to d(x, y).
20. 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.
(iii) Prove that Xis complete if and only if for every decreasing sequence F1⊇F2⊇ · · ·
of nonempty closed subsets of Xwith limn→∞ diam (Fn) = 0 we have
∞
\
n=1
Fn={a},for some a∈X.