Numerical Sequences and Series
Written by Men-Gen Tsai email: [email protected] 1. Prove that the convergence of cfw_sn implies convergence of cfw_|sn |. Is the converse true? Solution: Since cfw_sn is convergent, for any > 0, there exists N such tha
Dierentiation
Written by Men-Gen Tsai email: [email protected] 1. Let f be dened for all real x, and suppose that |f (x) f (y )| (x y )2 for all real x and y . Prove that f is constant.
) f Proof: |f (x) f (y )| (x y )2 for all real x and y . Fix y , |
The Riemann-Stieltjes Integral
Written by Men-Gen Tsai email: [email protected] 1. Suppose increases on [a, b], a x0 b, is continuous at x0 , f (x0 ) = 1, and f (x) = 0 if x = x0 . Prove that f R() and that f d = 0. Proof: Note that L(P, f, ) = 0 for a