MATH 410.501
Examination 2
March 26, 2014
Name:
ID#:
The exam consists of 3 questions. The point value for a question is written next to the
question number. There is a total of 100 points. No aids are permitted.
1. [50] In each of the following ten cases, indicate whether the given statement is true or
false. No justification is necessary.
(a) If x and y are nonzero vectors in Rn , then x · y = kxkkyk if and only if x = ty for
some real number t ≥ 0.
n
(b) Let {xk }∞
k=1 be a convergent sequence in R . Then the set {xk : k ∈ N} is not
closed.
1
n
∞
(c) Let {xk }∞
k=1 be a sequence in R such that the sequence {kxk k}k=1 converges in R.
∞
Then the sequence {xk }k=1 converges.
2
(d) The set {(x, y) ∈ R2 : 0 < y < ex } is open in R2 .
(e) Let n, m ∈ N and let T : Rn → Rm be a linear transformation. Then the operator
norm kT k := supkxk6=0 kT (x)||/kxk is finite.
(f) Let A be a subset of R2 whose boundary is empty. Then A must be either empty
or equal to all of R2 .
2
(g) Let A be a subset of Rn . Then its closure A is not open.
(h) The set {(x, y) ∈ R2 : y = x or y = 2x} is connected.
(i)
x3 + y 4
exists.
(x,y)→(0,0) x2 + 3y 4
lim
(j) Let A ⊆ Rn . Then the closure of the interior of A is equal to A.
3
2. [25] (a) State what it means for a set A ⊆ Rn to be compact.
(b) Prove that if A and B are compact subsets of Rn then so is A ∪ B.
(c) Give an example to show that the union of infinitely many compact subsets of R2
need not be compact. Provide justification.
4
3. [25] (a) Let n, m ∈ N. Let a be a vector in Rn , U an open subset of Rn containing a,
and f : U \ {a} → Rm a function. State what it means for f (x) to converge to a vector
L ∈ Rm as x → a.
(b) Prove that
lim
(x,y)→(0,0)
x2 y
, x2 + y 2
x2 + y 2
5
= (0, 0).