If the function f:D→ℝ is uniformly continuous and a is any number, show that the function a*f:D→ℝ also is uniformly continuous. Ok, so I am just learning my proofs so be patient with me, i'm very new at it. take a>0, ε>0 and x,y in D. We know |x-y|<δ whenever |f(x)-f(y)|<ε. If we take a*f:D→ℝ, we have |a*f(x)-a*f(y)|<ε → |a*[f(x)-f(y)]|<ε→ a*|f(x)-f(y)|<ε→ |f(x)-f(y)|<ε/a. Therefore, if we use ε/a, the result is proven. This just seems a little too easy to me, plus I've only done a few of these on my own. any suggestions/advice are greatly appreciated. Also, do I need to do this separately for a<0?