Description
Q. # 1
Unformatted Attachment Preview
3
1 - 13
37
{ol, [n, 6), (a, b)
(
(2,3
Cauchy seq. in lol
which fails to converge to an ele
is a
3
Tuesday
12
1. Find a set in (R, dı) that is neither open nor closed.
2. Show that ((0, 1), ds) is not complete
. finite a new
3. For each n e N let fn(x) = q”. Does {fn} converge in (C[0, 1], d2) (as in Example
1.4)?
4. Show that d, from Example 1.4 is not a metric for p E (0,1). (Hint: show that the triangle
inequality fails.)
(
15
sy
5. Show that a closed subset of a compact set is compact.
2.2.2
6. Use the fact that R is complete to show that Rd is complete. Follow the standard outline
for demonstrating completeness:
1) Take an arbitrary Cauchy sequence.
2) Find a candidate for the limit.
3) Prove that your candidate limit is in the space (this will be especially easy here).
4) Prove that your candidate is the limit.
eit 17. Show that a set E c Cd is bounded if and only if it is totally bounded. (Hint: Use the
fact that sets in R2d are bounded if and only if they are totally bounded.) Homomorphism
Thomerophism
8. Let Mn(F) be the set of n xn matrices with entries in F. Define d: Mn(F) ~ Mn(F) → R
by
d(A, B) = sup{||(A - B)x|| : x EF", ||3|| 00
کن (-1
2.1. 28
such that xnl
beunded
- lim
stupid math
19. Let (X, d) be a metric space. Show that d: X X X + R given by
d(x,y)
[0, 1]
d(x,y)
basic
1+ d(x, y)
is also a metric. (Hint: Use the properties of f(x)
In particular, you should
1+2
show that f(x + y) = f(x) + f(y) for x,y > 0.)
ball different
topology similar -
office your Monday
-
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Explanation & Answer
The proof is here.
1. Prove that the set [0, 1) in ℝ with the standard metric is neither open nor close.
We’ll use contradiction proof for both statements.
a) [0, 1) i...
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