let X be set ; :: thesis: for A being Subset of X
for A1 being SetSequence of X st A1 is convergent holds
( A1 (\) A is convergent & lim (A1 (\) A) = (lim A1) \ A )
let A be Subset of X; :: thesis: for A1 being SetSequence of X st A1 is convergent holds
( A1 (\) A is convergent & lim (A1 (\) A) = (lim A1) \ A )
let A1 be SetSequence of X; :: thesis: ( A1 is convergent implies ( A1 (\) A is convergent & lim (A1 (\) A) = (lim A1) \ A ) )
assume A1: A1 is convergent ; :: thesis: ( A1 (\) A is convergent & lim (A1 (\) A) = (lim A1) \ A )
A2: lim_inf (A1 (\) A) = (lim_inf A1) \ A by Th77
.= (lim A1) \ A by A1, KURATO_0:def 5 ;
then lim_sup (A1 (\) A) = lim_inf (A1 (\) A) by Th84;
hence ( A1 (\) A is convergent & lim (A1 (\) A) = (lim A1) \ A ) by A2, KURATO_0:def 5; :: thesis: verum