Multiply as sequence of number reverses the direction. If, for example, the sequence were increasing, say 1, 2, 3, 4, etc., then multiplying by -1 changes it to -1, -2, -3, -4, etc. which is decreasing.
The sequence 1/2, 0, 2/3, 0, 3/4, ..., does not converge but has one subsequence, 0, 0, 0, 0,..., that converges to 0 and another, 1/2, 2/3, 3/4, ..., that converges to 1. For this sequence liminf= 0 and limsup= 1. Multiplying each term by -1 gives -1/2, 0, -2/3, 0, -3/4, ... which has one sequence, 0, 0, 0, ..., that converges to 0 and another, -1/2, -2/3, -3/4, ..., that converges to -1. For this sequence, liminf= -1 and limsup= 0.
