Suppose $\displaystyle f(x)=\cos{x^2}-\cos{(x+1)^2}$.

(1)Find the upper and lower limit of $\displaystyle f(x) \text{ as }x\to +\infty$.

(2)Let A be the upper limit, B be the lower limit. prove or disprove that

for any real number c in the closed interval [B,A], there is a number sequence $\displaystyle \{x_n\} \text{ such that }x_n\to +\infty,f(x_n)\to c$.