of 21-355
Principles Real Analysis I Practice Problems
Fall 2004
Part I (Short Answer) 1. Consider the real sequence defined by xn = (-1)n + 1 2
n n
+ (-1)n +
1 n
for all n N.
Find lim supxn and lim inf xn .
n
2. Find the interior and closure of S if S = {-1} {x Q : 0 < x < 1} (1, 2]. 3. Give an example of an infinite set S R such that every subset of S is closed. 4. Determine whether or not f is uniformly continuous on S. sin(x3 ) for all x S. 1 + x2 1 (b) S = (0, 1), f (x) = for all x S. x x4 for all x S. (c) S = [0, 1], f (x) = 1+x (a) S = R, f (x) = 5. Give an example of a function f : R R such that f is differentiable at exactly one point. 6. Give an example of a countably infinite collection {Tn : n N} of subsets of R such that
cl(Tn ) = cl
n=1 n=1
Tn
.
7. Determine whether or not {fn } converges uniformly on S. n=1 (a) S = [0, ), (b) S = (0, 1), (c) S = (1, ), fn (x) = x n n n+x x n for all x S, n N. for all x S, n N. for all x S, n N. 1
fn (x) = sin2 fn (x) =
8. If f : R R is continuous and T is a closed subset of R, does it follow that f [T ] is closed. (Recall that f [T ] = {f (x) : x T }.) Explain. 9. If f : R R is twice differentiable and f (0) = f (1) = f (2) = 0, can we conclude that there exists z (0, 2) with f (z) = 0? Explain. 10. an Give example of two bounded sequences {xn } and {yn } such that n=1 n=1 lim inf (xn + yn ) > lim inf xn + lim inf yn
n n n
and lim sup(xn + yn ) <
n
lim sup xn
n
+ lim sup yn .
n
Part II (Give Complete Proofs.) 1. Let S, T be subsets of R. Show that cl(S T ) = (cl(S)) (cl(T )). 2. Let f : R R be given and put S = {x R : f (x) = 0}. Assume that f is differentiable on R and that f (x) = 0 for all x S. Define g : R R by g(x) = |f (x)| for all x R. Show that g is differentiable on R. 3. Let S R, M, > 0, and f, g : S R be given. Assume that |f (x)| M, |g(x)| x S and that f, g are uniformly continuous on S. Define F : S R by F (x) = f (x) x S. Show that F is uniformly continuous on S. g(x) 4. Assume that g : R R is uniformly continuous and let ...