1. The problem statement, all variables and given/known data Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A. 2. Relevant equations 3. The attempt at a solution F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where |x-u|<δ(ε) then |f(x)-f(u)|<ε. Since f is uniformaly continuous on A, then it is continuous at every point of A. Since A is a bounded subset, there exists m>0 st m > |x| for all x in A. Lastly, assume f is unbounded on A. Since f is unbounded on A, if given k >0 there exists a sequence (Xk) in A where |f(Xk)|>k for all k in ℝ. Since A is bdd, seq (Xk) is bdd. Since F is continuous at every point and A is bdd. then there exist a subequence of (Xk) denoted by (Xkn) that converges to x. Then it follows that F(Xkn) converges to F(x). Therefore F(Xkn) must be bounded. This is a contradiction a this would imply that there exists k0 where |F(Xkn)|≤ k0 while |F(Xkn)| should be ≥ k for all k in ℝ. **My proofs are always long winded like this** If at all possible, could somebody first, help me if i solved this problem wrong, and if i am right, show me a more concise way to write this problem up.