Does Scalar Multiplication Preserve the Supremum in Simple Sets?

[Seacow1988]

If S=sup {Sn: n>N}, is it true that kS= sup{kSn: n>N} where k is any scalar?

Thanks!

 

[Seacow1988]

Woops. I meant k is any nonnegative scalar.

Thanks!

 

[micromass]

Yes, I think that this is indeed true for nonnegative scalars k. The proof is not so hard: just show that S is an upper bound and that no lower upper bound exists.