Let A=event space. Assume that { A n } is a monotone nondecreasing sequence of events in A A. Let { A n } be a sequence of sets defined as follows: B 1 = A 1 ; B i = A i − A i − 1 for i ∈ { 2 , 3 , 4 , … }Prove that { B n } is a sequence of mutually exclusive events in A. Justify each line of proof.I know that what I need to show here is B i ∩ B j = ∅ for any i &lt; j. I've divided it to be proof by cases where case 1 is i = 1 and j ∈ { 2 , 3 , 4 , … } and case 2 is i ∈ { 2 , 3 , 4 , … } and j ∈ { 2 , 3 , 4 , … } where i &lt; j so that B i = A i − A i − 1 and B j = A j − A j − 1 and B i = A i − A i − 1 and B j = A j − A j − 1 where i ∈ { 2 , 3 , 4 … } and j ∈ { i + 1 , i + 2 , … }.

Question

Let   A=event space. Assume that   {      A    n    } is a monotone nondecreasing sequence of events in A  A. Let   {      A    n    } be a sequence of sets defined as follows:      B    1    =      A    1    ;      B    i    =      A    i    −      A          i      −      1           for     i  ∈  {  2  ,  3  ,  4  ,  …  }Prove that   {      B    n    } is a sequence of mutually exclusive events in   A. Justify each line of proof.I know that what I need to show here is       B    i    ∩      B    j    =  ∅ for any   i  &amp;lt;  j. I&#039;ve divided it to be proof by cases where case   1 is   i  =  1     and     j  ∈  {  2  ,  3  ,  4  ,  …  } and case   2 is   i  ∈  {  2  ,  3  ,  4  ,  …  } and   j  ∈  {  2  ,  3  ,  4  ,  …  } where   i  &amp;lt;  j so that       B    i    =      A    i    −      A          i      −      1           and         B    j    =      A    j    −      A          j      −      1       and       B    i    =      A    i    −      A          i      −      1           and         B    j    =      A    j    −      A          j      −      1       where   i  ∈  {  2  ,  3  ,  4  …  } and   j  ∈  {  i  +  1  ,  i  +  2  ,  …  }.

tykoyz · Accepted Answer

Your idea is correct. As a first step, take an element   e  ∈      B    j  . You have to prove that   e  ∉      B    i  . For this, use the definition       B    j    =      A    j    −      A          j      −      1      . Since   e  ∈      A    j    −      A          j      −      1      , what does this tell you about   e?

Kenya Leonard · Accepted Answer

If   i  &amp;lt;  j then       B    i    ⊆      A    i    ⊆      A          j      −      1       and       A          j      −      1        ∩      B    j    =  ∅.

Let A=event space. Assume that {A_n} is a monotone nondecreasing sequence of events in A. Let {A_n} be a sequence of sets defined as follows: B_1=A_1; B_i=A_i−A_(i−1) for i in {2,3,4,…} Prove that {B_n} is a sequence of mutually exclusive events in A. Justify each line of proof.

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