show-that-if-A-R-m-and-B-R-n-are-compact-sets-then-A-B-a-b-R-m-n-a-A-and-b-B-

Question Number 82792 by M±th+et£s last updated on 24/Feb/20

s h o w t h a t i f A \subset R^{m} a n d B \subset R^{n} a r e

c o m p a c t s e t s .

t h e n A \times B = {(a, b) \in R^{m + n} : a \in A a n d b \in B}

Commented by M±th+et£s last updated on 24/Feb/20

i s c o m p a c t

Answered by mind is power last updated on 24/Feb/20

s i n c e A i s a s u b s e t o f {I R}^{n} d i m {I R}^{n} = n

l e t N_{1} b e e a n o r m e i d o n t k n o w e n g l i s h n a m o f

n o r m e d e f i n i t i o n n N (x) = 0 \Leftrightarrow x = 0

N (x + y) ⩽ N (x) + N (y)

N (k x) =∣ k ∣ N (x)

l e t N_{2} n o r m e o f {I R}^{m}

N_{3} (x) = N_{1} (a) + N_{2} (b) i s n o r m e o v e r A \times B

x = (a, b) \in A \times B

s i n c e A i s c o m p a c t \forall U_{n} \in A \exists U_{φ (n)} \to l \in A

\Rightarrow N_{1} (U_{φ (n)} - l) \to 0

s i n c e B i s C o m p a c t \forall V_{n} \in B \exists l_{2} \in B a n d \exists V_{δ (n)} s u c h N_{2} (V_{δ (n)} - l_{2}) \to 0

w e c h e c k t h a t

\forall (U_{n}, V_{n}) \in A \times B (U_{φ (n)}, V_{δ (n)}) C v (l, l_{2})

N ((U_{φ (n)}, V_{δ (n)}); (l, l_{2})) = N_{1} (U_{φ (n)}, l) + N_{2} (V_{δ (n)}, l_{2}) \to 0

\Rightarrow \forall (U_{n}, V_{n}) \in A \times B h a s a c v s u b s e q u e n c e s \Rightarrow A \times B i s c o m p a c t s e t

Commented by M±th+et£s last updated on 24/Feb/20

n i c e s o l u t i o n s i r t h a n k y o u

Commented by mind is power last updated on 24/Feb/20

w i t h e p l e a s u r