Solution
= Solution
Separate off the integer part of the phase. Set $d_n=\sqrt{n^2+n}-n$. Rationalization gives
$$
d_n=\frac1{\sqrt{1+1/n}+1}\longrightarrow\frac12.
$$
Since squaring removes the sign in $\cos(\pi n+u)=(-1)^n\cos u$, the given <sequence> satisfies $b_n=\cos^2(\pi d_n)$. The <continuity> of the cosine now gives
$$
\boxed{\lim_{n\to\infty}b_n=\cos^2(\pi/2)=0.}
$$