Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Advanced undergraduate

Real analysis

Name: ______________________

Date: ______________________

You will learn to

Use the completeness axiom to prove the monotone convergence theorem, the Bolzano–Weierstrass theorem and the Cauchy criterion
Prove the intermediate value, extreme value and uniform continuity theorems for continuous functions on $[a, b]$
Distinguish pointwise from uniform convergence, prove that uniform limits of continuous functions are continuous, and recognise when limits, integrals and derivatives can be interchanged
Define the Riemann integral by upper and lower sums, prove that continuous functions are integrable and that the indicator function of $\Q$ is not
Prove both parts of the fundamental theorem of calculus
Explain, with Weierstrass's example, why continuity does not imply differentiability anywhere

1.
Which of these sequences converge (as $n \to \infty$ )? Choose all that apply.
- A. $a_n = (-1)^n / n$
- B. $a_n = (-1)^n$
- C. $a_n = n^{1/n}$
- D. $a_n = \sin n$
- E. $a_n = 1 + \frac12 + \cdots + \frac1n$
2.
Let $a_n = \dfrac{n}{n + 1}$ , which converges to $1$ . For $\varepsilon = \dfrac{1}{50}$ , what is the smallest $N$ such that $|a_n - 1| < \varepsilon$ for every $n \ge N$ ?
3.
Each sequence converges pointwise to $0$ on $[0, 1]$ . Which converge uniformly on $[0, 1]$ ? Choose all that apply.
- A. $f_n(x) = x^n / n$
- B. $f_n(x) = x^n(1 - x)$
- C. $f_n(x) = \dfrac{nx}{1 + n^2x^2}$
- D. $f_n(x) = \dfrac{\sin(n x)}{\sqrt n}$
- E. $f_n(x) = n x (1 - x^2)^n$
4.
The extreme value theorem says: a continuous function on a closed bounded interval $[a, b]$ attains a maximum. Which example shows that the word closed cannot be dropped?
- A. $f(x) = x$ on $(0, 1)$
- B. $f(x) = x$ on $[0, \infty)$
- C. $f(x) = x$ for $x < 1$ , $f(1) = 0$ , on $[0, 1]$
- D. $f(x) = 5$ on $(0, 1)$
5.
Here is a flawed proof that a pointwise limit $f$ of continuous functions $f_n$ is continuous at $c$ .
1. Let $\varepsilon > 0$ .
2. For each $x$ , choose $N_x$ with $|f_n(x) - f(x)| < \varepsilon/3$ for $n \ge N_x$ .
3. Let $N = N_c$ and choose $\delta > 0$ with $|f_N(x) - f_N(c)| < \varepsilon/3$ when $|x - c| < \delta$ .
4. For $|x - c| < \delta$ : $|f(x) - f(c)| \le |f(x) - f_N(x)| + |f_N(x) - f_N(c)| + |f_N(c) - f(c)|$ .
5. Each of the three terms is less than $\varepsilon/3$ , so $|f(x) - f(c)| < \varepsilon$ .
Which step is wrong?
- A.Step 2
- B.Step 3
- C.Step 4
- D.Step 5
6.
Which Lean statement says that f n converges to g uniformly on the set S (and not merely pointwise)?
- A.∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, ∀ x ∈ S, |f n x - g x| < ε
- B.∀ x ∈ S, ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, |f n x - g x| < ε
- C.∃ N : ℕ, ∀ ε > 0, ∀ n ≥ N, ∀ x ∈ S, |f n x - g x| < ε
- D.∀ ε > 0, ∀ n : ℕ, ∃ N ≥ n, ∀ x ∈ S, |f N x - g x| < ε
7.
Let $f(x) = x^3$ on $[0, 2]$ and let $P_n$ split $[0, 2]$ into $n$ equal pieces. What is the smallest $n$ for which $U(f, P_n) - L(f, P_n) \le \dfrac{1}{100}$ ?
8.
The proof of the intermediate value theorem can be turned into an algorithm by bisection (the nested-interval form of completeness). Write bisect(f, a, b, steps) which assumes f(a) <= 0 <= f(b) and returns the pair (lo, hi) after steps halvings: at each step let m = (lo + hi) / 2; if f(m) <= 0 replace lo by m, otherwise replace hi by m. The invariant f(lo) <= 0 <= f(hi) is kept, and hi - lo is halved each time.
Computational — Practice
9.
With the usual distance $|x - y|$ , in which of these sets does every Cauchy sequence converge to a point of the set? Choose all that apply.
- A. $\Q$
- B. $[0, 1]$
- C. $(0, 1)$
- D. $\Z$
- E. $\{1/n : n \ge 1\} \cup \{0\}$
10.
Let $f_n(x) = \dfrac{\sin(n^2 x)}{n}$ on $\R$ . Which statements are true? Choose all that apply.
- A. $f_n \to 0$ uniformly on $\R$ .
- B. $f_n'(0) \to 0$ .
- C. $\int_0^1 f_n \to 0$ .
- D.Theorem 11 applies to $(f_n)$ and shows the limit has derivative $\lim f_n'$ .
- E.The limit function is differentiable.

Answer key — Real analysis

1.
The convergent ones are $(-1)^n/n \to 0$ and $n^{1/n} \to 1$ . The sequence $(-1)^n$ has subsequences converging to $1$ and $-1$ ; $\sin n$ is dense in $[-1, 1]$ ; and the harmonic partial sums fail the Cauchy condition because $a_{2n} - a_n \ge \tfrac12$ for every $n$ .
2.
$|a_n - 1| = \frac{1}{n + 1}$ , which is decreasing in $n$ . The condition $\frac{1}{n+1} < \frac{1}{50}$ means $n + 1 > 50$ , i.e. $n \ge 50$ . So $N = 50$ : for $n = 49$ the distance is exactly $\frac1{50}$ , which is not strictly smaller.
3.
Compute or bound $\sup |f_n|$ : $x^n/n$ gives $1/n$ ; $x^n(1-x)$ gives at most $1/(n+1)$ ; $\sin(nx)/\sqrt n$ gives $1/\sqrt n$ — all uniform. For $nx/(1+n^2x^2)$ the value at $x = 1/n$ is always $\tfrac12$ , and $nx(1-x^2)^n$ is about $\sqrt n/e$ at $x = 1/\sqrt n$ , so these two do not converge uniformly.
4.
$f(x) = x$ on $(0, 1)$ is continuous on a bounded interval, has $\sup f = 1$ , and never attains it, because the endpoint $1$ is missing. In the proof, the sequence $x_n$ with $f(x_n) \to \sup f$ has a subsequence converging to $1$ , which is not in the domain: $(0, 1)$ is not sequentially compact.
5.
Step 5 claims $|f(x) - f_N(x)| < \varepsilon/3$ for all $x$ near $c$ , but step 2 only guarantees this for $n \ge N_x$ , and $N = N_c$ may be smaller than $N_x$ . With uniform convergence one $N$ works for all $x$ simultaneously, and the same proof becomes correct (Theorem 8). The functions $x^n$ on $[0, 1]$ , with $c = 1$ approached from the left, show the conclusion really fails.
6.
∀ ε > 0, ∃ N, ∀ n ≥ N, ∀ x ∈ S, |f n x - g x| < ε. Moving ∀ x ∈ S to the front gives pointwise convergence; moving ∃ N before ∀ ε makes the statement say the sequence is eventually equal to g; replacing ∀ n ≥ N by ∃ N ≥ n only controls a subsequence.
7.
For an increasing function, $M_i - m_i = f(x_i) - f(x_{i-1})$ , so $U - L = \frac{2}{n}\sum_{i=1}^n \big(f(x_i) - f(x_{i-1})\big) = \frac{2}{n}\big(f(2) - f(0)\big) = \frac{16}{n}$ . We need $\frac{16}{n} \le \frac{1}{100}$ , i.e. $n \ge 1600$ . The same telescoping argument proves that every monotone function is Riemann integrable.
8.
```
def bisect(f, a, b, steps):
    lo, hi = a, b
    for _ in range(steps):
        m = (lo + hi) / 2
        if f(m) <= 0:
            lo = m
        else:
            hi = m
    return lo, hi
```
The intervals $[lo_k, hi_k]$ are nested with lengths $(b - a)/2^k$ . In $\R$ (not in $\Q$ !) they shrink to a single point $c$ ; the $lo_k$ increase to $c$ and the $hi_k$ decrease to $c$ , so by continuity $f(c) = \lim f(lo_k) \le 0$ and $f(c) = \lim f(hi_k) \ge 0$ . This is a second proof of the intermediate value theorem. With floating point the loop is limited by machine precision after about $50$ halvings.
9.
A subset $A \subseteq \R$ is complete exactly when it is closed: every Cauchy sequence in $A$ converges in $\R$ by Theorem 4, and the limit lies in $A$ for every such sequence if and only if $A$ is closed. So $[0, 1]$ , $\Z$ and $\{1/n\} \cup \{0\}$ are complete; $\Q$ and $(0, 1)$ are not.
10.
$\sup |f_n| = 1/n \to 0$ , so $f_n \to 0$ uniformly and the integrals tend to $0$ . But $f_n'(0) = n \to \infty$ : the derivatives do not converge, so Theorem 11 does not apply, and $\lim f_n'$ does not exist even though the limit function $0$ is differentiable. Derivatives are the most fragile of the three limit interchanges.