## Transcribed Text

1. (6 points) Let (fn) be the sequence of functions defined on [0,5] by
for n = 1,2,
Show directly from the definition (meaning you should not use any
theorems we have proved about uniform convergence - theorems from earlier in the book are
fine though) that (fn) converges uniformly on [0,5].
2. (6 points) Let g be a bounded continuous function on (0,1) and define f on (0,1) by f(x) =
(1 - x2)g(x). Show f is uniformly continuous on (0,1).
3. (6 points) Let f be a function defined and continuous on all of R. Also let ao € IR and suppose
(an) is the sequence defined by an = f(an-1) when 72 > 0 (so as = f(ao), a2 = (f(ao)), and
so on). Suppose finally that the sequence (an) defined this way converges to a finite number
L. Show that f(L) = L.
4. (6 points) Let (S, d) be the metric space S = R² with the usual Euclidean metric d. Let E
be the set
E = {(x,y) € S : 1 < x² + y2 < 2}
Show E is connected. Hint: Try to show that E is path-connected.
rove or disprove THREE of the following statements, for five points each (please note that
ere are five options - the last two are on the next page):
5.
(5
points)
Let (gk) be a sequence of continuous functions on [0, 1] such that 2001 9k converges
uniformly on [0, 1]. Set
Mk = sup{(gk(x) : x € [0,1]}.
Then K1 Mk converges.
6. (5 points) Let f be a continuous function on (0,1). Then f achieves at least one of its two
possible extrema on its domain (i.e. either f achieves a minimal value, a maximal value or
both). Note that weak minima and maxima count in this case (so a constant function is not
a counterexample).
7. (5 points) Let f be a continuous function on [a, b] which is weakly increasing (recall that
this means f f(x) < f(y) whenever X < y), and let xo € (a,b]. Then the left-handed limit
lim
f(x) exists.

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