I'm having problems with the following proof: If $s_n$ and $t_n$ are sequences, then does $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$?
Is there a theorem that proves this? Is this even true? If not, is there a counter example I could use to show that it is not true?
Take a sequence which has liminf -1, and create a new sequence by putting the old sequence as the even terms, and zeros for the odd terms.
Take another sequence with liminf -2, and make a new sequence by putting the old sequence as odd terms and zeros on the even terms.
Both new sequences retain their old liminf, but the term wise product sequence is uniformly zero.