Could someone please check my work

Let S be a nonempty bounded subset of RR and let k E R . Define KS = {ks : s ( S}. Prove the following: 4. If k inf(k . S) . Also by definition 3.3.5 for the supremum, sup(S) is the least upper bound of S, which means that for all s E S , s 0 is NOT an upper bound of S. Therefore, there exists a number, s' E S such that s' > sup(S) - & by definition 3.3.5(b). Since k inf(k . S) and s' E S , then k . s' > inf(kS) . Combining the inequalities, k . s' 0. By Theorem 3.2.8, inf(k . S)