Could someone please check my work

Let S be a nonempty bounded subset of R and let k E R . Define KS = {ks : s E S}. Prove the following: 3. If k inf(S) . Since k 0 is NOT a lower bound of S. Therefore, there exists a number, s' E S such that s' 0 . By Theorem 3.2.8, k . inf( S)