SOLUTION OF QUIZ 2
MINGFENG ZHAO
February 15, 2013
1. [4 Points] Determine whether the following function has a limit as (x, y) → (0, 0).
f (x, y) =
lim
Proof. Claim I:
(x,y)→(0,0)
x4 + y 2
.
x4 + 2y 2
x4 + y 2
does not exist.
x4 + 2y 2
In fact, we know that
lim lim
x→0 y→0
x4 + y 2
x4 + 2y 2
=
=
lim lim
y→0 x→0
x4 + y 2
x4 + 2y 2
lim 1
x→0
1
=
lim
y2
2y 2
lim
1
2
=
x→0 y→0
x→0
x4
x4
=
=
Hence lim lim
lim
y→0
y→0
1
.
2
x4 + y 2
x4 + y 2
=
6
lim
lim
, which implies that
y→0 x→0 x4 + 2y 2
x4 + 2y 2
lim
(x,y)→(0,0)
x4 + y 2
does not
x4 + 2y 2
exist.
2. [6 Points] Let E be a subset of Rn . Show that if E is compact, then E is closed and bounded.
Proof. Case I: E = ∅. It is clear that ∅ is compact, closed and bounded.
1
2
MINGFENG ZHAO
Case II: E 6= ∅ is compact. Look at the family {Bk (0) : k ≥ 1}, then Bk (0) is open for all k ≥ 1
and
∞
[
Rn ⊂
Bk (0).
k=1
In particular, we know that {Bk (0) : k ≥ 1} is an open cover of E. Since E is compact, then there
exists finitely many k1 , · · · , km ∈ N such that
E⊂
m
[
Bki (0).
i=1
Let R = max {k1 , · · · , km } > 0, then we know that Bki (0) ⊂ BR (0) for all 1 ≤ i ≤ m, which
implies that E ⊂ BR (0), in particular, E is bounded.
On the other hand, we want to show that E is closed, it is equivalent to show that Rn \E is open.
In fact, for any x ∈ Rn \E and fix x, for any y ∈ E, since x 6= y, then there exists some ry > 0 and
Ry > 0 such that
Bry (x)
\
BRy (y) = ∅.
Notice that {BRy (y) : y ∈ E} is an open cover of E, since E is compact, there exists finitely many
y1 , · · · , yk such that
E⊂
k
[
BRyi (yi ).
i=1
Claim I: Let r = min {ry1 , · · · , ryk } > 0, then Br (x)
k
[
\
!
BRyi (yi )
= ∅, in particular, Br (x) ⊂
i=1
Rn \E.
If the Claim I is not true, then there exists some z ∈ Br (x)
\
k
[
i=1
there exists some 1 ≤ ii ≤ k such that
z ∈ Br (x)
\
BRyi (yi0 ).
0
!
BRyi (yi ) , which implies that
SOLUTION OF QUIZ 2
3
By the definition of r, r = min {ry1 , · · · , ryk } > 0, then Br (x) ⊂ Bryi (x). But by the construction
0
of Bryi (x), we know that Bryi (x)
0
0
\
BRyi (yi0 ) = ∅, which implies that Br (x)
0
\
BRyi (yi0 ) = ∅, we
0
get a contradiction. Therefore, we know that the Claim I is true.
By the Claim I, we know that for any x ∈ Rn \E, there exists some r > 0 such that Br (x) ⊂ Rn \E,
which implies that E is closed.
In a summary, we know that if E is compact in Rn , then E is closed and bounded.
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT
06269-3009
E-mail address: mingfeng.zhao@uconn.edu