Take a sequence a_n and define a new sequence b_n as b_n = inf{a_n, a_(n+1), a_(n+2), ...}. Then lim inf a_n is defined as lim b_n - that's the mathematical meaning of the term "lower limit." You're being asked to prove lim b_n = sup b_n. This can be done by proving b_n is nondecreasing.