Infinite Sequence of Unique Random Integers
Python 3, 137 127 123 138 117 99 98 bytes


  A named or unnamed function which returns an infinite iterable (such as a Python generator)


from random import*
def f(x=[],y=1):
 while y in x or.5&lt;random():y+=1
 yield y;yield from f(x+[y])



-10 bytes by putting the conditional in one line (Thanks, @Dennis)
-4 bytes by initializing r at the beginning of the loop
+15 bytes by remembering numbers can contain zeroes...
-21 bytes by renaming randint, and moving the condition in the head of the inner loop
-18 bytes by switching to a functional paradigm
-1 bytes by moving from or ...&gt;.5 to .5&lt;random() (thanks, @JonathanAllan)