# How to prove that $\sup (c \cdot X) = c \cdot \sup X$? [closed]

Let $c$ be a positive real number and let $X$ be a bounded from above, non-empty subset of $\mathbb R$. Define set $c \cdot X$ as $$c \cdot X = \{ c \cdot x \bracevert x \in X \} .$$ Prove that $c \cdot X$ is non-empty and bounded from above and that $\sup (c \cdot X) = c \cdot \sup X$.

## closed as off-topic by user99914, Davide Giraudo, Claude Leibovici, user91500, Frits VeermanMay 16 '17 at 9:47

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• Where are you stuck? – florence May 16 '17 at 8:22
• In fact don't really know how to start. I'm a high school student so it's not really my level of maths. Just started attending extra math lessons and got some exercises to start with. I've read a lot and understood what the question is about, what supremum is etc., but haven't seen many mathematical proofs before and don't really know how it's supposed to look like and be formulated. – MrLeatherpants May 16 '17 at 17:36

Hint. If $c>0$ then $x\leq y$ iff $cx\leq cy$.