\displaystyle \limsup_{n \to \infty} \left[ \frac{1}{n}, 5 + \frac{1}{n} \right] = \bigcap_{n = 1}^{\infty} \bigcup_{k = n}^{\infty} \left[ \frac{1}{k}, 5 + \frac{1}{k} \right] = \bigcap_{n = 1}^{\infty} \left\langle 0, 5 + \frac{1}{n} \right] = \left\langle 0, 5 \right]