The Real and Complex Number Systems
Written by Men-Gen Tsai email: [email protected] 1. 2. 3. 4. 5. 6. Fix b > 1. (a) If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that (bm )1/n = (bp )1/q . Hence it makes sense to define br =
Basic Topology
Written by Men-Gen Tsai email: [email protected] 1. Prove that the empty set is a subset of every set. Proof: For any element x of the empty set, x is also an element of every set since x does not exist. Hence, the empty set is a su
Numerical Sequences and Series
Written by Men-Gen Tsai email: [email protected] 1. Prove that the convergence of {sn } implies convergence of {|sn |}. Is the converse true? that |sn - s| < Solution: Since {sn } is convergent, for any > 0, there ex
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
a b
f d = 0.
Proof: Note that L(P
Differentiation
Written by Men-Gen Tsai email: [email protected] 1. Let f be defined 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. Proof: |f (x) - f (y)| (x - y)2 for all real x and 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
a b
f d = 0.
Proof: Note that L(P
Solutions to Exam 1
1. Let A Z. Since (a) = 0, it follows that inf n (En ) = 0, where the inf is taken collections of elementary sets {En }n>0 , s.t. n En is an open cover of A. Thus for any m > 0, there is an open cover of elementary sets {En }n>0
The Pigeonhole Principle
Rosen 4.2
Pigeonhole Principle
If k+1 or more objects are placed into k boxes, then there is
at least one box containing two or more objects.
Generalized Pigeonhole Principle
If N objects are placed into k boxes, then there is at