Number of integers not exceeding $n$? Is it valid to say that the number of positive integers not exceeding some integer $n$ is $n$? I can't think what else it would be but that seems deceptively simple.
 A: That is correct. You can prove it by induction (lol). 
The number of positive integers less than or equal to $1$ is $1$ so we're good for $n=1$.
Then assume true for $n$, i.e. "there are $n$ distinct positive integers $\leq n$"
Now we must prove true for $n+1$
$n+1$ must have $1$ more distinct positive integer which is $\leq n+1$ than $n$. Therefore there are $n+1$ positive integers $\leq n+1$ as required. 
A: It never hurts to doublecheck assumptions like that.
However, your assumption is true only if $n$ itself is a positive integer. Maybe you intended to write "some positive integer $n$" but didn't for whatever reason.
If $n < 1$, then the number of positive integers not exceeding $n$ is 0.
But if $n > 0$, then you are indeed correct: the number of positive integers not exceeding some integer $n$ is $n$ itself.
How many positive integers are greater than or equal to 1? Only 1 itself. How many positive integers are greater than or equal to 2? Just 1 and 2. How many positive integers are greater than or equal to 3? 1, 2 and 3. And so on and so forth. The positive integers count themselves.
