Prove that if k > 0 and (s_n) is a bounded sequence, lim sup ks_n = k * lim sup s_n and what happens if k<0?

My attempt: If (s_n) is bounded, then lim sup s_n = S, where S is a real number, so lim sup (ks_n)= lim sup(k)(s_n)=S*lim sup(k) = S*k = k*lim sup s_n.

Let lim inf s_n = L. If k<0, then lim sup (ks_n) = -lim inf(-k)(s_n)= -L*lim inf (-k) = -k *- L = k * lim inf (s_n).

Is this correct?