Could someone please check my work

Let S be a nonempty bounded subset of R 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' 2 sup(S) - & by definition 3.3.5(b). Since k 0. By Theorem 3.2.8, inf(k . S)