Elementary Analysis. Homework 6

MATH5301 Elementary Analysis. Homework 6. 

6.1
(a) Show that for any x > 0, x ∈ R, limn→∞
x
1/n = 1.
(b) Show that for any bounded sequence {an} and any sequence {bn}, converging to zero, the sequence {anbn}
converges to zero.
(c) Find the limit limn→∞
an where an =
s
2 + r
2 + q
2 + · · · +
√
2
| {z }
n
. Prove the convergence.
(d) Find the limit limn→∞
an where an = 2 +
1
2 +
1
.
.
. +
1
2
. Prove the convergence.
6.2 True or False?
(a) If f is continuous function (S1, d1) 7→ (S2, d2) and U ⊂ S1 is open then f(U) ⊂ S2 is also open?
(b) If f is continuous function (S1, d1) 7→ (S2, d2) if and only if for any closed set C ⊂ S2 the set f
−1
(C) ⊂ S1
is also closed.
(c) If f and g are continuous functions (S, d) 7→ R, then m(x) := max(f(x), g(x)) and n(x) = min(f(x), g(x))
are also continuous?
(d) If f is continuous function (S, d) 7→ R then for any Cauchy sequence {xn} in S, the sequence f(xn) is
Cauchy sequence in R.
6.3
Prove the following properties of continuous functions:
(a) For any a, b ∈ R and for any two continuous functions f, g : S1 7→ S2, it follows
(af + bg)(x) := af(x) + bg(x) : S1 7→ S2
is also continuous.
(b) For any continuous f : S1 7→ S2 and for any continuous h : S1 7→ R the function
(hf)(x) := h(x) · f(x) : S1 7→ S2
is also continuous.
(c) If h(x) 6= 0 for any x ∈ S1 then 1
h(x)
is also continuous function from S1 to R.
6.4
Prove the following statement: If A and B are two closed nonempty disjoint sets in the metric space (S, d) then
there exists a continuous function χ(x) such that χ(x) = 0 for all x ∈ A and χ(x) = 1 for all x ∈ B.
Hint:
(a) Define the distance from the point x to the set A as
ρA(x) := inf
y∈A
d(x, y)
(b) Show that ρA(x) = 0 ⇔ x ∈ A¯
(c) Show that ρA(x) is Lipshits with constant 1.
(d) Consider χ(x) = ρB(x)
ρA(x) + ρB(x)
.
6.5
Which of the following sets in R
2 are compact?
(a) A = {(x, y) | x
2 − y
2 6 1}
(b) B = {(x, y) | 0 < x2 + y
2 6 1}
(c) C = {(x, y) | x
2 + y
4 6 1}
(d) D = {(1,
1
n
) | n ∈ N} ∪ (1, 0)
6.6
Let A ⊂ S be compact set. Show that
(a) ∂A is compact.
(b) For any closed B, A ∩ B is compact.
(c) For any compact C, A ∪ C is compact.
(d) Union of infinitely many compact sets may be not compact.