Real Analysis Test 1

Front 1

Well-Ordering Property of N 

Back 1

Every nonempty subset of N has a least element.

Front 2

Uniqueness Theorem

Back 2

If S is a finite set, then the number of elements in S is a unique number in N.

Front 3

Cantor's Theorem

Back 3

If A is any set, then there is no surjection on to the set P(A) of all subsets of A.

Front 4

Triangle Inequality

Back 4

If a,b in R, then |a+b| <= |a|+|b|.

Front 5

The Completeness Property of R

Back 5

Every nonempty set of real numbers that has an upper bound also has a supremum in R.

Front 6

Archimedean Property

Back 6

If x in R, then there exists n_x in N such that x<= n_x

Front 7

Density Theorem

Back 7

If x and y are any real numbers with x < y, then there exists a rational number r in Q such that x < r < y.

Front 8

Characterization Theorem

Back 8

If S is a subset of R that contains at least two points and has the property:

      if x,y in S and x<y, then [x,y] in S

then S is an interval.

Front 9

Nested Intervals Property

Back 9

If I_n = [a_n, b_n], n in N, is a nested sequence of closed bounded intervals, then there exists a number E in R such that E in I_n for all n in N. 

Front 10

Squeeze Theorem

Back 10

Suppose that X=x_n, Y=y_n, and Z=z_n are sequences of real numbers such that

      x_n <= y_n <= z_n

and that lim(x_n) = lim(z_n). Then Y = y_n is convergent and 

lim(x_n) = lim(y_n) = lim(z_n)

Front 11

Monotone Convergence Theorem

Back 11

A monotone sequence of real numbers is convergent iff it is bounded. If it is increasing, it converges to its supremum. If it is decreasing it converges to its infimum.