Real analysis

Which of these sequences converge (as $n \to \infty$)? Choose all that apply.

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 $n \ge N$?

Each sequence converges pointwise to $0$ on $[0, 1]$. Which converge uniformly on $[0, 1]$? Choose all that apply.

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?

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?

Which Lean statement says that f n converges to g uniformly on the set S (and not merely pointwise)?

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 $U(f, P_n) - L(f, P_n) \le \dfrac{1}{100}$?

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.

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.

Let $f_n(x) = \dfrac{\sin(n^2 x)}{n}$ on $\R$. Which statements are true? Choose all that apply.