4. The notation $[x]$ represents the greatest integer not exceeding $x$ (for example $[\sqrt{5}]$ $=2)$. Let $n$ be a natural number, and
$I=(n+1)^{2}+n-\left[\sqrt{(n+1)^{2}+n+1}\right]^{2}$, then
(A) $I>0$;
(B) $I<0$;
(C) $I=0$;
(D) When $n$ takes different values, all three cases may occur.
Answer $(\ldots)$
4. (A)

Solve $\because n$ is a positive integer,
$\therefore$ there is the equation
$$
\begin{aligned}
& {\left[\sqrt{(n+1)^{2}+(n+1)}\right]^{2}=[\sqrt{(n+1)+(n+2)}]^{2} } \\
= & (n+1)^{2} \text { holds. }
\end{aligned}
$$

Therefore, $\quad I=(n+1)^{2}+n-(n+1)^{2}=n>0$.
$\therefore$ (A) is correct.