Analysis practice

Some practice on ideas from Analysis.

Simply pick the answer you think is correct. You'll get immediate feedback. Whether you're right or not, you can keep choosing answers. When you're ready, move on to the next question.

Let (x_n) be a sequence of real numbers. Suppose the sequence is decreasing and bounded below by 0. What is the most we can deduce from this?
1. The sequence tends to 0 as n tends to infinity.
2. The sequence converges as n tends to infinity.
3. The sequence converges to a non-negative number as n tends to infinity.
4. We cannot tell whether the sequence converges as n tends to infinity.
Let (x_n) be a sequence of real numbers. What does it mean to say that x_n tends to a limit x as n tends to infinity?
1. There is some N so that if n ≥ N then |x_n - x| < ε.
2. If n is big enough, then x_n is extremely close to x (in fact, as close as we like).
3. For any ε > 0, there is some N (which may depend on ε) such that if n ≥ N then |x_n - x| < ε.
4. There is some N so that for any ε > 0, if n ≥ N then |x_n - x| < ε.
Let (x_n) be a sequence of real numbers. What does it mean to say that x_n does not tend to a limit as n tends to infinity?
1. For any x and any ε > 0, there is N so that if n ≥ N then |x_n - x| > ε.
2. For any x there is some ε > 0 and some N so that if n ≥ N then |x_n - x| > ε.
3. For any x there is some ε > 0 so that for any N if n ≥ N then |x_n - x| > ε.
4. For any x there is some ε > 0 such that for any N there is some n ≥ N with |x_n - x| > ε.
Let f be a function from R to R. What does it mean to say that f is continuous at the point a?
1. For any ε > 0, there is some δ > 0 so that if |x - a| < δ then |f(x) - f(a)| < ε.
2. For any ε > 0, there is some δ > 0 so that if |x - a| < ε then |f(x) - f(a)| < δ.
3. There is some δ > 0 so that if |x - a| < δ then |f(x) - f(a)| < ε.
4. There is some ε > 0 so that for any δ > 0, if |x - a| < δ then |f(x) - f(a)| < ε.
Let x_n be a sequence of real numbers. Here are some possible implications about convergence of infinite sums. The sums are all over n from 1 to infinity unless otherwise indicated. Tick any of these implications that are always true.

For each one that's true, can you give a proof? For each one that isn't always true, can you give a counterexample?
1. If Σ |x_n| converges, then Σ x_n converges.
2. If Σ x_n converges, then Σ |x_n| converges.
3. If x_n tends to 0 as n tends to infinity, then Σ x_n converges.
4. If Σ x_n converges, then x_n tends to 0 as n tends to infinity.
5. If the sum of x_n from n = 100 to infinity converges, then Σ x_n converges.
6. If Σ x_n converges, then the sum of x_n from n = 100 to infinity converges.
7. Let S_N be the sum of x_n from n = 1 to N. If the sequence S_N converges as N tends to infinity, then Σ x_n converges.
8. If Σ (x_n)/n converges, then Σ x_n converges.
9. If (y_n) is a sequence with 0 ≤ x_n ≤ y_n for all n and Σ y_n converges, then Σ x_n converges.
Let f be a function from R to R. What does it mean to say that f is not continuous at a?
1. For any ε > 0, there is δ > 0 so that if |x - a| < δ then |f(x) - f(a)| ≥ ε.
2. There is some ε > 0 so that for any δ > 0, there is some x with |x - a| < δ and |f(x) - f(a)| ≥ ε.
3. There is some ε > 0 so that for any δ > 0, if |x - a| < δ then |f(x) - f(a)| ≥ ε.
4. For any ε > 0, there is δ > 0 and there is some x with |x - a| < δ and |f(x) - f(a)| ≥ ε.
Let (x_n) be an infinite sequence of real numbers. Which of the following implications are true? Tick any that are true.

For each statement that's true, can you give a proof? For each that is not always true, can you give a counterexample?
1. If x_n tends to x, then (x_n)² tends to x².
2. If (x_n)² tends to x², then x_n tends to x.
3. If x_n tends to x, then any subsequence of (x_n) also tends to x.
4. If a subsequence of (x_n) tends to x, then x_n tends to x.
5. If x_n ≤ 10 for all n, then x_n tends to a limit that is at most 10.
6. If x_n ≥ 0 for all n and x_n is a decreasing sequence, then x_n tends to a limit.
7. If there's an M so that |x_n| ≤ M for all n, then (x_n) has a convergent subsequence.
8. If there's an M so that |x_n| ≤ M for all n, then any convergent subsequence of (x_n) must tend to the same limit.
Let f be a continuous function from the interval [a,b] to the real numbers. What is the most we can deduce about f?
1. f is bounded.
2. f is bounded and attains its bounds.
3. f need not be bounded.
4. f is bounded but need not attain its bounds.
Let f be a function from R to R. What does it mean to say that f(y) tends to l as y tends to x?
1. There is ε > 0 so that for any δ > 0, if 0 < |y - x| < δ then |f(y) - l| < ε.
2. There is ε > 0 and there is δ > 0 so that if 0 < |y - x| < δ then |f(y) - l| < ε.
3. For any ε > 0, there is δ > 0 so that if 0 < |y - x| < δ then |f(y) - l| < ε.
4. For any ε > 0 and any δ > 0, if 0 < |y - x| < δ then |f(y) - l| < ε.
Let f be a function from [0,1] to R, and suppose that it is twice differentiable on [0,1]. Assume that f(0) =0. What can you deduce about f? Tick any that apply.

For each true statement, give a proof. If the statement need not be true, give a counterexample.
1. If f(1) = 0, then there is some point t in (0,1) with f ' (t) = 0.
2. If f(1) = 0, then for any t in (0,1), we have f ' (t) ≥ 0.
3. If f ' (t) > 0 for all t, then f(t) > 0 for all t in (0,1).
4. If f ' (t) ≥ 0 for all t, and f(1) = 0, then f(t) = 0 for all t in (0,1).
5. If f(1) = 0 and f '' (t) ≥ 0 for all t in (0,1), then f(t) ≤ 0 for all t in (0,1).
6. If f(1) = 1, then there is some t in (0,1) with f ' (t) = 1.
7. If f(1) = 1, then f ' (t) ≥ 0 for all t in (0,1).