Let D be updated by:
where d is bounded by 
, 
,
and the initial value of
.
Then:
Proof:
The highest D can be is if it is always updated with :
so 
.
Similarly
.
I should note this only works if
α
is between 0 and 1.
Proof: In the discrete case, Q is updated by:
so by Theorem B.1:
This can also be viewed in terms of temporal discounting:
Similarly:
For example, if 
, then 
.
And (assuming 
) as 
, 
.
Note that since 
, it follows that 
.
Proof: In the discrete case, W is updated by:
so by Theorem B.1:
by Theorem B.2.
Similarly:
Note that since , it follows that 
.
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