September 24, 2001
Math 323 — Practice Exam I
1. (25 points) Using the definition of limit (i.e., not the Algebraic Limit Theorem), prove the
following:
(a) lim(n/√n2+ 1) = 1.
(b) If lim xn=xand lim yn=y, then lim(xn+ 2yn) = x+ 2y.
2. (20 points) In class, we proved (in an unnecessarily complicated way) that, if an≥bnfor all
n∈Nand lim an=a∗and lim bn=b∗, then a∗≥b∗.
(a) Show by example that we cannot replace ≥with >throughout the statement.
(b) Prove on the basis of the result above that, if an≥Bfor all n∈N(where Bis a real
constant), then lim an≥B.
3. (20 points) Consider the function
f(x) =
−1 if x< 0,
0 if x= 0 ,
1 if x> 0.
For a set Afor which inf A= 0, is it true that inf{f(a) : a∈A}=f(inf A)? What can
we say about sets which also have infima, but the infima are different from 0? What is the
corresponding statement for suprema?
4. (10 points) Archimedes proposed finding the circumference of a circle as follows: “Inscribe in
the circle an equilateral triangle, then double the number of sides to form a regular hexagon,
then again to form a regular 12-sided figure (a dodecagon), and so on. The perimeters of
these polygons get close to the circumference of the circle.” Put aside the question of whether
this process gives the correct answer. What do we need to show to be sure that it gives any
answer at all? And how might we show it?
