Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Since $\sqrt{n^2+1}$ is an algebraic integer for $n=1001,\dots,2000$ and algebraic integers are closed under addition, it follows that $$S=\sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$$ is algebraic integer too. Now use the fact that any rational algebraic integer IS an integer. This contradicts the fact that $S$ is NOT an integer (as you have already shown). Hence $S$ is not rational.

A root of a monic polynomial with integer coefficients is called an algebraic integer. For example, $\sqrt{n}$ is an algebraic integer, because it is a root of $X^2 - n$. An important result is that the algebraic integers form a ring, i.e. the sum and product of algebraic integers is an algebraic integer. See here for my favourite proof of this fact.