First number is greater, here is the explanation:
for - (99^999 + 1)/(99^1000 + 1)
consider it as (99^999 + 1)/99^1000 , 99^1000 is big enough that adding 1 does not make a lot of difference
(99^999 + 1)/99^1000= (99^99/99^1000)+(1/99^1000)= (1/99)+1/99^1000 ......(1)
for-(99^1000 +1)/(99^1001 + 1)
consider it as (99^1000 +1)/99^1001
(99^1000 +1)/99^1001=(1/99)+(1/99^1001) ..........(2)
on comparing 1 and 2 we get (99^999 + 1)/(99^1000 + 1) as bigger one.