

* indicates the problem will be collected 

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Chapter 1:Analysis WebNotes by John Lindsay Orr Introduction

1. Make a list of 10 theorems that you feel are critical to
understanding and using the material you learned in the first year of
calculus. Be sure to state clearly the theorem with all its hypotheses and conclusions. 2. For each of the 10 theorems you gave, provide examples of how the theorem can fail if only one of the hypotheses is not satisfied. 

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*1.Give a statement and proof of the Mean value Theorem and
of Rolle's Theorem. [You may use other results from the top 10 for your
proof. If you use Rolle's theorem to prove the MVT prove Rolle's
Theorem without the MVT!] *2. Prove: Rolle's Theorem is equivalent to the Mean Value Theorem. *3. For any real number x let x be the absolute value of x, i.e., x = x when x>0,x =x when x< 0, and x = 0 when x =0. Prove: For any real numbers a and b, a+b < or = a + b. *4. Prove the following: (i) ab < or = a + b. (ii) a  b < or = ab. (iii) ( a  b ) < or = ab *5. Suppose m(x,y) = yx for x and y real numbers.Prove that for any real numbers, a,b, and c: (i) m(a,b) = 0 if and only if a=b. (ii) m(a,b) = m (b,a). (iii) m(a,b) < or = m(a,c) + m(b,c) (iv) If m(a,b) = m(a,c) + m(b,c) then either a= c, b=c, a<c< b, or b<c<a. 

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*1.Give a statement the Extreme Value Theorem with examples illustrating how it can be false if one of the hypotheses is not satisfied. *2. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form `[a_j, b_j]` with `a_j < b_j` for `j` = 1 to n and set of numbers `{ c_k: k = 1 `to` m}`. Prove that there are numbers p and q ` in D`, where `f(p) <= f(x) <= f(q)` for all `x in D`. *3. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form `[a_j, b_j]` with `a_j < b_j` for `j `= 1 to n and set of numbers `{ c_k: k = 1` to `m}`. Using the intermediate value theorem, prove that f (D) = {y : y = f (x) for some x in D} is also the union of a finite number of intervals of the form `[r_j, s_j]` with `r_j < s_j` for `j = 1` to ` n` and set of numbers `{ t_k: k = 1` to ` m}`. 

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1. Axioms for the Real numbers2. Sequences of Numbers 
*1. Suppose A and B are nonempty sets of numbers and that for any x in A and any y in B, `x <=y`. a) Prove that the `LUB (A) <= y` for all y in B b) Prove that `LUB(A) <= GLB( B)`. *2. Suppose A and B are nonempty sets of numbers that are bounded above. Let A+B denote the set of numbers that can be expressed as a+b where a is in A and b is in B. Prove: `LUB (A + B) = LUB(A) + LUB(B)` *3 .a) Suppose `I_1 = [a_1,b_1], I_2 = [a_2,b_2], ... `is a sequence of close intervals with `a_n<b_n`, `a_n<=a_{n+1)` and ` b_n >= b_{n+1}` for all `n`. Prove: There is a number c that is in all the intervals. b) Give a counterexample to show that the statement in part a is not true if the intervals are open instead of closed. 

1013 
1. Let f and g be functions which are continuous on the whole of R and with f (0) = g (0). a.* Prove that the function defined by h(x) = f(x) for x 0 and h(x) = g(x) for x > 0 is continuous everywhere. b.* Prove that the absolute value function  x  is continuous everywhere. c.* If f is a continuous function on R, prove that the function  f(x)  is also continuous on R. d.* If f and g are continuous functions prove that the function M(x) = max{f (x), g (x)} is also continuous. [Hint: Prove that max(a, b) =((a + b) + a  b)/2 for any real numbers a and b.] e. Prove that the function m(x) = min{f (x), g (x)} is continuous if f and g are. 2.a.* Use the geometry of the sine function on the unit circle to prove that  sin x  sin y   x  y  b. * Prove that the function f(x) = sin x is continuous everywhere. c. Prove that the function cos x is continuous everywhere. 3.* Suppose f :[0,1] `>` [0,1] is a continuous function. Prove there is some `a in [0,1]` where `f(a) = a`. 4. Prove that if a subsequence of a Cauchy sequence converges, then the original Cauchy sequence converges. 5. a. Prove that ` 1/{n+1} < ln(n+1) ln(n) < 1/ n `. b. Let `a_n = 1 + 1/2 + 1/ 3 + ... +1/n  ln(n)`. Show that `a_n >= 0` and `a_n > a_{n+1}` for all `n`. [thus there is a limit for the `a_n` which is called Euler's constant and denoted `gamma`.] 
