Question Bank - Mathematics for Economists

Question Bank: Mathematics for Economists --- Instructor: <a href="mailto:gopakumar@igidr.ac.in">Gopakumar Achuthankutty</a> Teaching Assistants: <a href="mailto:abhijeet.m@igidr.ac.in">Abhijeet Kumar Mishra</a> and <a href="mailto:24phdeco01@igidr.ac.in">Gaurav Patel</a> --- ### Module 1 1. How do you formally define the tail of a sequence? Then, a sequence $\{s_{n}\}$ converges if and only if the tail of the sequence $\{s_{n}\}$ converges. 2. If a sequence $\{s_{n}\}$ diverges to $\infty$ then the sequence $\{t_{n}\}$ where for all $n \in \mathbb{N}$, $t_{n} = \dfrac{s_{n}}{s_{n}+1}$, converges. 3. Using ideas from how we defined sequences that diverge to $+\infty$ in class, how can you define a sequence that ==diverges to $-\infty$==? Let $\{s_{n}\}$ and $\{t_{n}\}$ be sequences that diverge to $-\infty$. Prove the following statements: (i) The sequence $\{s_{n}+t_{n}\}$ diverges to $-\infty$. (ii) The sequence $\{s_{n}\cdot t_{n}\}$ diverges to $\infty$. (iii) If $\forall n \in \mathbb{N}$, $s_{n} \neq 0$ then the sequence $\left\{\dfrac{1}{s_{n}}\right\}$ converges to $0$. 4. Let $\{s_{n}\}$ be a bounded sequence. Further, let $$\underline{x}=\inf \left\{s_{n} \mid n \in \mathbb{N}\right\} \text { and } \overline{x}=\sup \left\{s_{n} \mid n \in \mathbb{N}\right\} .$$ Suppose that, moreover, for all $n \in \mathbb{N}$, $\underline{x}<s_{n}<\overline{x}$. Prove that there is a pair of convergent subsequences $\{t_{k}\}$ and $\{r_{k}\}$ so that the sequence $\{\lvert t_{k}-r_{k} \rvert\}$ converges to $\underline{x}-\overline{x}$. 5. Let $a,b \in \mathbb{R}$ such that $a<b$. If $f:\left[a,b\right] \rightarrow \left[a,b\right]$ is a continuous function then show that there exists a point $x^{*} \in \left[a,b\right]$ such that $f(x^{*})=x^{*}$. 6. Let $X \subseteq \mathbb{R}$. A function $f:X \rightarrow \mathbb{R}$ is called ==Hölder continuous== if $\exists M,\alpha > 0$, $\forall x,y \in X$, $\lvert f(x)-f(y) \rvert < M \lvert x-y \rvert^{\alpha}$. Show that if a function $f:X \rightarrow \mathbb{R}$ is Hölder continuous then it is uniformly continuous. 7. Determine necessary and sufficient conditions on a pair of sets $A$ and $B$ so that they will have the property that there exists a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)=0$ for all $x \in A$ and $f(x)=1$ for all $x \in B$. 8. A student proved the statement "*Every bounded differentiable function $f: \mathbb{R} \to \mathbb{R}$ is constant*" as follows: *By assumption, there exist real numbers $M, N$ such that $$N \leq f(x) \leq M.$$ Taking derivatives, we get $$0 \leq f^{\prime}(x) \leq 0.$$ Hence, $f^{\prime}(x)=0$, implying that $f$ is constant.* Is this proof correct? Justify your answer. 9. A function $f: \left(a,b\right) \to \mathbb{R}$ has a symmetric derivative at a point $x \in \left(a,b\right)$ if $$f_s^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}$$ exists. Show that $f_s^{\prime}(x)=f^{\prime}(x)$ at any point $x \in \left(a,b\right)$ at which the latter exists but that $f_s^{\prime}(x)$ may exist even when $f$ is not differentiable at $x$. 10. Let $f: \left(a,b\right)$ be a continuously differentiable function. Then, for all $x \in \left(a,b\right)$, the derivative $f'(x)$ is bounded if and only if $f$ is Lipschitz continuous. 11. If $f$ is infinitely differentiable on $[a, b]$ and $n \geq 1$, show that $$f(b)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(b-a)^k+\frac{1}{n!} \int_a^b f^{(n+1)}(x)(b-x)^n dx.$$ 12. Let $\left\{a_0, a_1, \ldots\right\}$ be an increasing sequence in $[a, b]$ such that $a_0=a$ and $a_n \rightarrow b$. If $f$ is a bounded function on $[a, b]$ that is integrable on $\left[a_n, a_{n+1}\right]$ for all $n \geq 0$, is $f$ integrable on $[a, b]$?