On page 68 of the fourth edition of Tao's Analysis 1, is Exercise $3.5.12$, the first part of which I believe is erroneous. The exercise is stated as follows:
(Note: $n++$ refers to the successor of $n$)
Let $X$ be a set, let $f:\mathbf{N}\times X\to X$ be a function, and let $c$ be an element of $X$. Show that there exists a function $a:X\to X$ such that
$$a(0) = c$$
and
$$a(n++) = f(n,a(n)) \text{ for all }n\in\mathbf{N},$$
and furthermore that this function is unique.
The required properties of the function $a$ imply that the natural numbers lie in its domain. However, the domain specified here is any arbitrary set $X$ which may not contain the natural numbers. If the domain of $a$ was restricted to the set of natural numbers, the statement of the exercise itself serves as a complete definition of the function, from which it's existence and uniqueness follow directly (I may be wrong here. Do correct me if that's the case). However, this exercise is followed by a fairly elaborate hint, which indicates that the solution may not be this trivial.
Alternatively, if $X$ was defined to be some set that contains the natural numbers but isn't limited to them, one could define $a$ as in the exercise statement for the natural numbers and as some constant for all elements of $X$ that aren't natural numbers, but then there would be no guarantee of uniqueness, as there would be more than one choice of this constant.
Thus, it seems to me that this exercise is somewhat incorrect, but I would like to get some more opinions on this. Any views you may have on this would be much appreciated.
Update: This error has now been reported, acknowledged, and listed as an erratum on the book's official site which can be found here.
Best Answer
I believe that it is probably a typo. The following is the exercise on page $68$ of Analysis $1$, third edition. The set $X$ is the set of natural numbers $N$.