Could someone please check my work

Let S be a nonempty bounded subset of IR and let k E R . Define k5 = {kms 6 S}. Prove the following: 4. If 3k 0 is NOT an upper bound of S by definition 3.3.5. Then sup(S) - 8 sup(S) e by definition 3.3.5(b). Since k- 0. By Theorem 3.2.8, inf(k . S) s k - sup(S)- Since k ' sup(S) S inflk ' S) and inUC ' S) _<_ k sup . then infucs therefore if inf>