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A[n], B[n] are sequences of subsets of X. A,B c X.

Given that lim A[n] = A and lim B[n] = B

show that

lim ( A[n] \ B[n] ) = A \ B.

Given that lim A[n] = A and lim B[n] = B

show that

lim ( A[n] \ B[n] ) = A \ B.

Use:

(1) lim B[n] = B ==> lim X\B[n] = X\B

(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B

(1) lim B[n] = B ==> lim X\B[n] = X\B

(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B

A[n], B[n] are sequences of subsets of X. A,B c X.

Given that lim A[n] = A and lim B[n] = B

show that

lim ( A[n] + B[n] ) = A + B.

Given that lim A[n] = A and lim B[n] = B

show that

lim ( A[n] + B[n] ) = A + B.

Use:

(1) lim B[n] = B ==> lim X\B[n] = X\B

(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B

(1) lim B[n] = B ==> lim X\B[n] = X\B

(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B

What does it mean that a complex function satisfies the Cauchy-
Riemann equations at a point in its domain?

z:-A ; A c |C ; A is open ; f:A->|C ; U = Re(f) ; V = Im(f)

(1) all four partial derivatives exist at z: Ux, Uy, Vx, Vy

(2) Ux = Vy and Uy = -Vx

page 18 in OLDTIMER

(1) all four partial derivatives exist at z: Ux, Uy, Vx, Vy

(2) Ux = Vy and Uy = -Vx

page 18 in OLDTIMER

Suppose that a complex function is defined on an open set and
satisfies the Cauchy-Riemann equations at a point. Does it have
to be diffable at that point?

f(x,y) = sqrt( |x*y| ) at (0,0)

C-R satisfied, not diffable

C-R satisfied, not diffable

lim (x,y)->(0,0) [ x*y / sqrt( x*x + y*y) ] = ?

= 0

What is separate continuity?

Let (X,d) , (Y,g) be two metric spaces.

Consider f:X*Y->|R.

(1) /\y:-Y f is continuous as a function of x:-X

(2) /\x:-X f is continuous as a function of y:-Y

(1)and(2) <=> separate continuity

Consider f:X*Y->|R.

(1) /\y:-Y f is continuous as a function of x:-X

(2) /\x:-X f is continuous as a function of y:-Y

(1)and(2) <=> separate continuity

(X,d) is a metric space; a:-X; f:X->|R; f(x)=d(x,a).

Show that f is Lipschitz.

Show that f is Lipschitz.

Use the second triangle inequality.

What is a directed set?

(X,>)

X is a set and > is a binary relation on X such that

(1) /\a,b:-X \/ c:-X [ c>a and c>b ]

(2) /\a,b,c:-X [ a>b and b>c ==> a>c ]

X is a set and > is a binary relation on X such that

(1) /\a,b:-X \/ c:-X [ c>a and c>b ]

(2) /\a,b,c:-X [ a>b and b>c ==> a>c ]

What is a net?

h is a net contained in X

means that

there is a directed set (T,>) (see the previous item)

h:T->X

means that

there is a directed set (T,>) (see the previous item)

h:T->X

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

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

Prove that (T,>) is a directed set.

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

Prove that (T,>) is a directed set.

Very easy.