If C 1, CZ, C3 . are sets such that Cr C Ch+ 1 . K = 1, 2, 3 . limk_ too CR is defined as the union C , UC 2 UC , U . Find lim k_ too CK. ( 2 ) CK =...

Please help with Probability Space functions. The problem is shown in the attachment. I was struggling in understand the concept and I'll be grateful if you can show me the steps to solutions. Thanks in advance!

1 . If C 1, CZ, C3 . .... are sets such that Cr C Ch+ 1 . K = 1, 2, 3 . .... limk_ too CR is defined as theunion C , UC 2 UC , U .... Find lim k_ too CK.( 2 ) CK = Lac ER : 1 / K < < < 3 - 1/ Kj , K = 1, 2, 3 , ... .( b ) Ck = ( ( 2 , 4 ) ER 2 : 1 / k < 2 2 + y 2 { 1 - 1 / K} , K = 1, 2, 3 . ... .2 . If C 1 , C2, C3 . .... are sets such that ( * ) Ck+1 . K = 1 , 2 , 3 . ... . limk_ too CR is defined as theintersection CIn Cz N C; M .... Find limk_ too CK.( a ) CK. = {` ER : 2 - 1 / K < < < 2) , K = 1, 2, 3 . ....( b ) Ck = { ER : 2 < < < 2 + 1 / K; ] , K = 1 , 2, 3 , ... .( C ) Ck = ( ( 2 , 4 ) E RR 2 : 0 < 2 2 + y ? < 1 / K;] , K = 1, 2, 3 . ....3. For every one- dimensional set C , define the function @ ( C ) = > of (a ) , where f (a ) - 12/ 31 ( 1/ 3 )2 .I = 0. 1, 2. .... zero elsewhere . If ( 1 = 12 : 2 - 0. 1, 2, 31 and C 2 - 12 : 2 - 0, 1, 2.... ], findQ ( ( 1 ) and Q ( ( 2 ) . Hint : Recall that S = at art ... + arn- 1 = a ( 1 - pro ) / ( 1 - 8) , and hence , itfollows that Jim ~ _ too S = a / ( 1 - 7 ) provided that | ~| < 1 .4. Let C be a set in one- dimensional space and let @ ( C ) be equal to the number of points in C' whichcorrespond to positive integers . Then Q ( C ) is a function of the set C . Find @ ( C ) .( 2 ) C = PIER : 0 < < < 5).( b ) C = 6 - 2, - 1} .( C ) C = LIER : - 00 < < < 6}.