第一部份:高等微積分

1. (10%)Suppose that a_n>=0　, a_n+m<=a_n+a_m for all m,n N . Prove that the limit exists.
   
   
2. A real-valued function f defined on (a,b) is said to be convex if
   　　　
   whenever x,y (a,b) and λ [0,1].

   (a)	(10%)Prove that every convex function defined on (a,b) is continuous on (a,b).
   (b)	(15%)Assume that g is a real-valued function defined on (a,b) such that 　　　 for all x,y (a,b). Is g a convex function on (a,b)?

3. Define
   　　　　

   (a)	(8%)Letγ be a differentiable mapping from R1 into R2, with γ(0)=(0,0), and . Define g(t)=f(γ(t)) for t R. Prove that g is differentiable at t = 0.
   (b)	(7%)Prove that　f　is not differentiable at (0,0)

　第二部分:實變函數論

1. (10%)Let (X,Ω,μ) be a measure space with . If ,
   prove that .
   
   
2. (10%)for , define
   　　　　　　(nZ)
   Show that, if for all n Z, then f = 0 a.e.
3. (15%)Let (X,Ω,μ) be a measure space. Suppose that , for n=1,2,....Prove that , if and , then there exists a subsequence { f_nj } of { f_n } such that .
   
   
4. (15%)Suppose that Ω is a σ-algebra of sets in a set X. Prove that Ω is either finite or uncountable.
   
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