Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

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

lim_inf (A[n] u B[n]) c lim_inf (A[n]) u lim_inf (B[n])

Is this true?

lim_inf (A[n] u B[n]) c lim_inf (A[n]) u lim_inf (B[n])

Is this true?

No.

Put A[1] = {1}, A[2] = {2}, A[3] = {1}, A[4] = {2}, etc.

Put B[1] = {2}, B[2] = {1}, B[3] = {2}, B[4] = {1}, etc.

The right side is empty.

Put A[1] = {1}, A[2] = {2}, A[3] = {1}, A[4] = {2}, etc.

Put B[1] = {2}, B[2] = {1}, B[3] = {2}, B[4] = {1}, etc.

The right side is empty.

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

lim_inf (A[n]) u lim_inf (B[n]) c lim_inf (A[n] u B[n])

Is this true?

lim_inf (A[n]) u lim_inf (B[n]) c lim_inf (A[n] u B[n])

Is this true?

Yes.

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

lim_sup ( A[n] u B[n] ) c ( lim_sup A[n] ) u ( lim_sup B[n] )

Is this true?

lim_sup ( A[n] u B[n] ) c ( lim_sup A[n] ) u ( lim_sup B[n] )

Is this true?

Yes.

Even equality holds.

lim_sup ( A[n] u B[n] ) = ( lim_sup A[n] ) u ( lim_sup B[n] )

Even equality holds.

lim_sup ( A[n] u B[n] ) = ( lim_sup A[n] ) u ( lim_sup B[n] )

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

( lim_sup A[n] ) u ( lim_sup B[n] ) c lim_sup ( A[n] u B[n] )

Is this true?

( lim_sup A[n] ) u ( lim_sup B[n] ) c lim_sup ( A[n] u B[n] )

Is this true?

Yes.

Even equality holds.

( lim_sup A[n] ) u ( lim_sup B[n] ) = lim_sup ( A[n] u B[n] )

Even equality holds.

( lim_sup A[n] ) u ( lim_sup B[n] ) = lim_sup ( A[n] u B[n] )

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] u B[n] ) = A u B.

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

show that

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

A u B c

( lim_inf A[n] ) u ( lim_inf B[n] ) c

lim_inf ( A[n] u B[n] ) c

lim_sup ( A[n] u B[n] ) c

( lim_sup A[n] ) u ( lim_sup B[n] ) c

A u B

( lim_inf A[n] ) u ( lim_inf B[n] ) c

lim_inf ( A[n] u B[n] ) c

lim_sup ( A[n] u B[n] ) c

( lim_sup A[n] ) u ( lim_sup B[n] ) c

A u B

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

( lim_inf A[n] ) n ( lim_inf B[n] ) c lim_inf ( A[n] n B[n] )

Is this true?

( lim_inf A[n] ) n ( lim_inf B[n] ) c lim_inf ( A[n] n B[n] )

Is this true?

Yes.

Even equality holds.

( lim_inf A[n] ) n ( lim_inf B[n] ) = lim_inf ( A[n] n B[n] )

Even equality holds.

( lim_inf A[n] ) n ( lim_inf B[n] ) = lim_inf ( A[n] n B[n] )

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

lim_sup ( A[n] n B[n] ) c ( lim_sup A[n] ) n ( lim_sup B[n] )

Is this true?

lim_sup ( A[n] n B[n] ) c ( lim_sup A[n] ) n ( lim_sup B[n] )

Is this true?

Yes.

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

( lim_sup A[n] ) n ( lim_sup B[n] ) c lim_sup ( A[n] n B[n] )

Is this true?

( lim_sup A[n] ) n ( lim_sup B[n] ) c lim_sup ( A[n] n B[n] )

Is this true?

No.

A[2*n] = {1}; A[2*n+1] = {2}

B[2*n] = {2}; B[2*n+1] = {1}

A[2*n] = {1}; A[2*n+1] = {2}

B[2*n] = {2}; B[2*n+1] = {1}

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] n B[n] ) = A n B.

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

show that

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

Use

(1) ( lim_inf A[n] ) n ( lim_inf B[n] ) c lim_inf ( A[n] n B[n] )

(2) lim_sup ( A[n] n B[n] ) c ( lim_sup A[n] ) n ( lim_sup B[n] )

(1) ( lim_inf A[n] ) n ( lim_inf B[n] ) c lim_inf ( A[n] n B[n] )

(2) lim_sup ( A[n] n B[n] ) c ( lim_sup A[n] ) n ( lim_sup B[n] )

B[n] is sequences of subsets of X. B c X.

Given that lim B[n] = B

show that lim X\B[n] = X\B.

Given that lim B[n] = B

show that lim X\B[n] = X\B.

page 19 in 1st measure