2. Given A R, we let L be the set of limit points of A and dene the closure of A to be
A = A [ L.
(a) Show that x is a limit point for A if and only if for any ” > 0, there exists y 2
A (x “; x + “) such that y 6= x.
(b) Show that L is closed.
Hint: it may be easier using part (a) than the denition in terms of sequences from
class.
(c) Let A;B R and let LA, LB denote the set of limit points for A and B, resp. Show
that the set of limit points for A [ B is contained in LA [ LB
(d) Use the results of part (b) and (c) to show that A [ L has the same limit points as
A. This shows that A = A.
(e) Show that the closure of A is the smallest closed set containing A i.e. for any closed
set F containing A, A F.
Note that, in general, A 6 L, i.e. not every point in A is necessarily a limit point of
A. As an example, take A = [0; 1] [ f10g. Then, 10 is not a limit point of A because
only the constant sequence xn = 10 converges to 10, but we explicitly do not all for the
sequences to equal the supposed limit point for every n 2 N.
3. Let’s do some more problems on closed sets.
(a) Let a; b 2 R. Prove that the interval [a; b] := fx 2 R : a x bg is a closed set.
(b) Let a < b. Prove that the interval (a; b) := fx 2 R : a < x < bg is not a closed set.
(c) Is Q a closed set? What is its set of limit points?
(d) Can a closed set in R be unbounded?
(e) Show that every closed and bounded subset F R has a maximum and a minimum.
(f) If fFng1n
=1 are closed sets in R, is 1 n=1Fn closed? Is [1 n=1Fn closed?
4. Prove using the “; denition of the functional limit that limx!2(x2 + 3x + 1) = 11.
6. Determine, if they exist, limx!0 f(x), limx!0+ f(x), limx!0 f(x), limx!1 f(x) and
limx! 1 f(x) for the following functions. Also, indicate when they do not exist.
(a) f(x) = x
jxj
(b) f(x) = 1
x
(c) f(x) =
p
1+3×2 1
x2
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