Question

Prove directly from definition that the set {1/2^n | n=1, 2, 3, ...} is not compact but {0, 1/2^n |n=1, 2, 3, ...} is compact.

Answer #1

prove directly by definition that [1, \infty) is not
compact.

Prove directly from the definition of the limit
(b) lim (n−2)/(n+12)=1
c) lim n√8 = 1. (Hint: recall the formula for x^n − 1).

Prove directly (using only the definition of the countably
infinite set, without the use of any theo-rems) that the union of a
finite set and a countably infinite set is countably infinite.

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Prove that lim x^2 = c^2 as x approaches c by appealing directly
to the definition of a limit.

Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.

Prove that in R^n with the usual topology, if a set is
closed and bounded then it is compact.

prove that a compact set is closed using the Heine - Borel
theorem

Problem 1. Suppose E is a given set,
and On, for n ∈ N, is
the set defined by
On = {x ∈ Rd :
d(x, E) < 1/n }.
(a) Prove that On is open.
(b) Prove that if E is compact, then m(E) =
limn →∞ m(On).
(c) Would the above be true for E closed and unbounded
set?
(d) Would the above be true for E open and bounded
set?

Prove by induction that 1*1! + 2*2! + 3*3! +... + n*n! = (n+1)!
- 1 for positive integer n.

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