1) Suppose that f:R^n->R is continuous, and f(u) >= norm(u) for every
point u in R^n.  Prove that f^(-1)([0,1]) is compact.  Note f^(-1)(A)
refers to the pre-image of f on A.

2) Let A and B be compact subsets of R.  Define K = {(x,y) in R^2 : x
in A, y in B}.  Prove that K is compact.

3) A mapping F:R^n->R^m is said to be Lipschitz if there is a number C
such that norm(F(u)-F(v)) <= C*norm(u-v), for all points u and v in
R^n.

The number C is called a Lipschitz constant for the mapping.  Show
that a Lipschitz mapping is uniformly continuous.

4) Is the product of two real-valued uniformly continuous functions
again uniformly continuous?

5) Show that the set S = {(x,y) in R^2 : either x or y is rational} is
pathwise connected.

6) Let u be a fixed point in R^n, and let c be a fixed real number. 
Prove that each of the following three sets in convex:

i)   {v in R^n : <v,u> > c}
ii)  {v in R^n : <v,u> > c}
iii) {v in R^n : <v,u> > c}

Note that <a,b> means the dot product (scalar product) of a with b.

7) Given a point u in R^n and a point v in R^m, we define the point
(u,v) to be the point in R^(n+m) whose first n components coincide
with components of u and whose last m components coincide with those
of v.  Suppose that A is a subset of R^n and that F:A->R^m is
continuous.  The graph G of this mapping is defined by

G = {(u,v) in R^(n+m) : u in A, v = F(u)}.

Show that if A is pathwise connected, then G is also pathwise connected.