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Let T be the set of all finite subsets of |N.

For a,b:-T, a>b <=> b is contained in a

(T,>) is a directed set.

Let t[n] = (-1)^(n+1) * (1/n), for n:-|N.

For F:-T, let S(F) = the finite sum of t[n] where n:-F.

Does the net S(F) converge?

For a,b:-T, a>b <=> b is contained in a

(T,>) is a directed set.

Let t[n] = (-1)^(n+1) * (1/n), for n:-|N.

For F:-T, let S(F) = the finite sum of t[n] where n:-F.

Does the net S(F) converge?

NO.

page 169 in "Topological Spaces, From Distance to Neighborhood"

page 169 in "Topological Spaces, From Distance to Neighborhood"

What is a singleton?

A singleton is a set that has exactly one element.

{x}

{x}

What do you call a set that has exactly one element?

a singleton ['siN g.l t.n]

Consider the complex function Re(z).

Prove that it is nowhere diffable.

Prove that it is nowhere diffable.

=

a,b:-[-pi,pi] ; cos(a)=cos(b)

Show that a=b or a=(-b).

Show that a=b or a=(-b).

hint: cos is 1-1 on [0,pi] and cos(|a|) = cos(|b|)

Let t[n] be a sequence satisfying -pi<=t[n]<=pi. Let -pi<t<pi.
Suppose that cos(t[n])->cos(t) and sin(t[n])->sin(t). What can
we conclude?

t[n]->t

page 38 in OLDTIMER

page 38 in OLDTIMER

Express differently:

max(x,y) = ???

max(x,y) = ???

max(x,y) = ( x + y + |x - y| ) / 2

Express differently

min(x,y) = ???

min(x,y) = ???

min(x,y) = ( x + y - |x - y| ) / 2

Let A[n] be a sequence of sets contained in X.

Prove that the two conditions are equivalent.

(1) A[n] converges

(2) /\x:-X [ {n:-|N : x:-A[n]} is finite or {n:-|N : not x:- A[n]} is finite} ]

Prove that the two conditions are equivalent.

(1) A[n] converges

(2) /\x:-X [ {n:-|N : x:-A[n]} is finite or {n:-|N : not x:- A[n]} is finite} ]

page 20 first measure theory

Let A[n] be a sequence of sets contained in X. Let A c X.

Prove that the three conditions are equivalent.

(1) A[n] -> A

(2) A[n] + A -> O

(3) lim_sup (A[n] + A) = O

Prove that the three conditions are equivalent.

(1) A[n] -> A

(2) A[n] + A -> O

(3) lim_sup (A[n] + A) = O

page 21 measure theory