Exercise 1 . Let *1, $2 . ... and / 1, 12 . ... be sequences of random variables such that In * * and In ~ {' both almost surely . Prove that In + In * * + \\' almost surely , and In In - *'Y' almost surely . Exercise 2 . Let *1 , *2 . ... and /1 , 12 . ... be sequences of random variables such that In - X and Y' -> > both in probability .* ( 1 ) Prove In + In * * * * in probability . ( 2 ) Prove *, In * *'Y' in probablety . Exercise 3 . Let *1, *2 . ... be independent random variables that are uniformly dis - tributed in \\ - 1 , 1 ] . Show that the sequence Y 1 , 1 2 . ... converges in probability to some* limit , and identify the limit , for each of the following cases :" ( 1 ) In = In / n;] ( 2 ) In = Am` ( 3 ) I'm = * 1 . $ 2 . .. In';` ( 4 ) In = max( ^ ! . .... In ). Exercise 1 . In problem 2 of this week's homework , you will show that for any 1 5 p