Real Analysis

Prove that $(\cos \theta )^{p} \leq \cos (p\theta )$ for $0 \leq \theta \leq \pi /2$ and $0 < p < 1$.

Let $f: [0,1] \rightarrow \mathbb {R}$ be continuously differentiable, with $f(0) = 0$. Prove that

Let $f(x)$ be a real valued function defined for all $x \geq 1$, satisfying $f(1) = 1$ and

Prove that

exists and is less than $1 + \frac{\pi }{4}$.

Let $f, g: [0,1] \rightarrow [0,\infty )$ be continuous functions satisfying

Prove that there exists $t \in [0,1]$ with $f(t)^{2}+3f(t) = g(t)^{2}+3g(t)$.

For $f$ a real valued function on the real line, define the function $\Delta f$ by $\Delta f(x) = f(x+1)-f(x)$. For $n \geq 2$, define $\Delta^{n} f$ recursively by $\Delta^{n} f = \Delta (\Delta^{n-1} f)$. Prove that $\Delta^{n} f = 0$ if and only if $f$ has the form $f(x) = a_{0}(x)+a_{1}(x)x+\cdots +a_{n-1}(x)x^{n-1}$ where $a_{0}, a_{1}, \ldots , a_{n-1}$ are periodic functions of period 1.

Suppose that $f: \mathbb {R} \rightarrow \mathbb {R}$ is a nonconstant function such that $f(x) \leq f(y)$ whenever $x \leq y$. Prove that there exist $a \in \mathbb {R}$ and $c > 0$ such that $f(a+x)-f(a-x) \geq cx$ for all $x \in [0,1]$.

Either prove or disprove (by a counterexample) each of the following statements:

Prove that for all $x > 0$, $\sin x > x - \frac{x^{3}}{6}$.

Let

Justify the statement

where

Suppose that $f$ is a twice differentiable real function such that $f''(x) > 0$ for all $x \in [a,b]$. Find all numbers $c \in [a,b]$ at which the area between the graph $y = f(x)$, the tangent to the graph at $(c, f(c))$, and the lines $x = a$, $x = b$, attains its minimum value.

Find the maximum area of all triangles that can be inscribed in an ellipse with semiaxes $a$ and $b$, and describe the triangles that have maximum area.

Note: See also Problem 2.2.2.

Let $f$ be a continuous real valued function on $[0,\infty )$. Let $A$ be the set of real numbers $a$ that can be expressed as $a = \lim_{n \rightarrow \infty } f(x_{n})$ for some sequence $(x_{n})$ in $[0,\infty )$ such that $\lim_{n \rightarrow \infty } x_{n} = \infty$. Prove that if $A$ contains the two numbers $a$ and $b$, then $A$ contains the entire interval with endpoints $a$ and $b$.

Show that the equation

has, for each sufficiently small $\varepsilon > 0$, exactly two solutions. Let $x(\varepsilon )$ be the smaller one. Show that

Suppose that $f(x)$ is a polynomial with real coefficients and $a$ is a real number with $f(a) \neq 0$. Show that there exists a real polynomial $g(x)$ such that if we define $p$ by $p(x) = f(x)g(x)$, we have $p(a) = 1$, $p'(a) = 0$, and $p''(a) = 0$.

Let $p(z)$ be a nonconstant polynomial with real coefficients such that for some real number $a$, $p(a) \neq 0$ but $p'(a) = p''(a) = 0$. Prove that the equation $p(z) = 0$ has a nonreal root.

Let $f$ be a $C^{2}$ function on the real line. Assume $f$ is bounded with bounded second derivative. Let

Prove that

Find all pairs of integers $a$ and $b$ satisfying $0 < a < b$ and $a^{b} = b^{a}$.

For which positive numbers $a$ and $b$, with $a > 1$, does the equation $\log_{a} x = x^{b}$ have a positive solution for $x$?

Which number is larger, $\pi^{3}$ or $3^{\pi }$?

For which numbers $a$ in $(1,\infty )$ is it true that $x^{a} \leq a^{x}$ for all $x$ in $(1,\infty )$?

Show that a positive constant $t$ can satisfy

if and only if $t < e$.

Suppose that $f(x)$ is defined on $[-1,1]$, and that $f'''(x)$ is continuous. Show that the series

converges.

If $f$ is a $C^{2}$ function on an open interval, prove that

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous. Suppose that $\mathbb {R}$ contains a countably infinite subset $S$ such that

if $p$ and $q$ are not in $S$. Prove that $f$ is identically 0.

Let the function $f$ from $[0,1]$ to $[0,1]$ have the following properties:

• $f$ is of class $C^{1}$;

• $f(0) = f(1) = 0$;

• $f'$ is nonincreasing (i.e., $f$ is concave).

Prove that the arclength of the graph of $f$ does not exceed 3.

Let $f$ be a real valued $C^{1}$ function on $[0,\infty )$ such that the improper integral $\int_{1}^{\infty } \left|f'(x)\right|\, dx$ converges. Prove that the infinite series $\sum_{n=1}^{\infty } f(n)$ converges if and only if the integral $\int_{1}^{\infty } f(x)\, dx$ converges.

Let $E$ be the set of all continuous real valued functions $u: [0,1] \rightarrow \mathbb {R}$ satisfying

Let $\varphi : E \rightarrow \mathbb {R}$ be defined by

Show that $\varphi$ achieves its maximum value at some element of $E$.

Let $S$ be the set of all real $C^{1}$ functions $f$ on $[0,1]$ such that $f(0) = 0$ and

Define

Show that the function $J$ is bounded on $S$, and compute its supremum. Is there a function $f_{0} \in S$ at which $J$ attains its maximum value? If so, what is $f_{0}$?

Let $f$ be a real valued continuous nonnegative function on $[0,1]$ such that

for $t \in [0,1]$. Show that $f(t) \leq 1+t$ for $t \in [0,1]$.

Suppose $\varphi$ is a $C^{1}$ function on $\mathbb {R}$ such that

Prove or give a counterexample: $b$ must be zero.

Show that

$0 \leq k < 1$, is an increasing function of $k$.

Given that

find $f'(t)$ explicitly, where

Define

Compute $F'(0)$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a nonzero $C^{\infty }$ function such that $f(x)f(y) = f\left(\sqrt{x^{2}+y^{2}}\right)$ for all $x$ and $y$ and that $f(x) \rightarrow 0$ as $\left|x\right| \rightarrow \infty$.

Let $S$ be the set of continuous real-valued functions on $[0,1]$ such that $f(x)$ is rational whenever $x$ is rational. Prove that $S$ is uncountable.

Let $f$ be a bounded, continuous, real-valued function on $\mathbb {R}^{2}$. Define the function $g$ on $\mathbb {R}$ by

Prove that $g$ is continuous.

Suppose that $f$ maps the compact interval $I$ into itself and that

for all $x, y \in I$, $x \neq y$. Can one conclude that there is some constant $M < 1$ such that, for all $x, y \in I$,

Let the real valued function $f$ on $[0,1]$ have the following two properties:

• If $[a,b] \subset [0,1]$, then $f([a,b])$ contains the interval with endpoints $f(a)$ and $f(b)$ (i.e., $f$ has the Intermediate Value Property).

• For each $c \in \mathbb {R}$, the set $f^{-1}(c)$ is closed.

Prove that $f$ is continuous.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be uniformly continuous with $f(0) = 0$. Prove: there exists a positive number $B$ such that $\left|f(x)\right| \leq 1+B\left|x\right|$, for all $x$.

Suppose that $f$ is a continuous function on $\mathbb {R}$ which is periodic with period 1, i.e., $f(x+1) = f(x)$. Show:

Let $h: [0,1) \rightarrow \mathbb {R}$ be a function defined on the half-open interval $[0,1)$. Prove that if $h$ is uniformly continuous, there exists a unique continuous function $g: [0,1] \rightarrow \mathbb {R}$ such that $g(x) = h(x)$ for all $x \in [0,1)$.

Let $f$ be a continuous real valued function on $[0,\infty )$ such that $\lim_{x \rightarrow \infty } f(x)$ exists (finitely). Prove that $f$ is uniformly continuous.

Prove or supply a counterexample: If the function $f$ from $\mathbb {R}$ to $\mathbb {R}$ has both a left limit and a right limit at each point of $\mathbb {R}$, then the set of discontinuities of $f$ is, at most, countable.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ satisfy $f(x) \leq f(y)$ for $x \leq y$. Prove that the set where $f$ is not continuous is finite or countably infinite.

A function $f: [0,1] \rightarrow \mathbb {R}$ is said to be upper semicontinuous if given $x \in [0,1]$ and $\varepsilon > 0$, there exists a $\delta > 0$ such that if $\left|y-x\right| < \delta$, then $f(y) < f(x)+\varepsilon$. Prove that an upper semicontinuous function $f$ on $[0,1]$ is bounded above and attains its maximum value at some point $p \in [0,1]$.

Prove that a continuous function from $\mathbb {R}$ to $\mathbb {R}$ which maps open sets to open sets must be monotonic.

Let $f$ be a continuous function from $\mathbb {R}$ to $\mathbb {R}$ such that $\left|f(x)-f(y)\right| \geq \left|x-y\right|$ for all $x$ and $y$. Prove that the range of $f$ is all of $\mathbb {R}$.

Note: See also Problem 2.1.8.

Let $f$ be a continuous function on $[0,1]$. Evaluate the following limits.

Let $f$ be a function from $[0,1]$ into itself whose graph

is a closed subset of the unit square. Prove that $f$ is continuous.

Note: See also Problem 2.1.2.

Let $f$ be a continuous real valued function defined on $[0,1] \times [0,1]$. Let the function $g$ on $[0,1]$ be defined by

Prove that $g$ is continuous.

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ be bounded on bounded sets and have the property that $f^{-1}(K)$ is closed whenever $K$ is compact. Prove $f$ is continuous.

Let $A_{1} \geq A_{2} \geq \cdots \geq A_{k} \geq 0$. Evaluate

Note: See also Problem 5.1.11.

Compute

Let $x_{0} = 1$ and

Prove that $x_{\infty } = \lim_{n \rightarrow \infty } x_{n}$ exists, and find its value.

Define a sequence of real numbers $(x_{n})$ by

Show that $(x_{n})$ converges, and evaluate its limit.

Let $\alpha$ be a number in $(0,1)$. Prove that any sequence $(x_{n})$ of real numbers satisfying the recurrence relation

has a limit, and find an expression for the limit in terms of $\alpha$, $x_{0}$ and $x_{1}$.

Let $k$ be a positive integer. Determine those real numbers $c$ for which every sequence $(x_{n})$ of real numbers satisfying the recurrence relation

has period $k$ (i.e., $x_{n+k} = x_{n}$ for all $n$).

Let $a$ be a positive real number. Define a sequence $(x_{n})$ by

Find a necessary and sufficient condition on $a$ in order that a finite limit $\lim_{n \rightarrow \infty } x_{n}$ should exist.

Let $x_{n}$ be a sequence of real numbers so that

Show that $\lim_{n \rightarrow \infty } x_{n} = x$.

Let $a$ and $x_{0}$ be positive numbers, and define the sequence $(x_{n})_{n=1}^{\infty }$ recursively by

Prove that this sequence converges, and find its limit.

Let $x_{1}$ be a real number, $0 < x_{1} < 1$, and define a sequence by $x_{n+1} = x_{n}-x_{n}^{n+1}$. Show that $\liminf_{n \rightarrow \infty } x_{n} > 0$.

Let $f(x) = \frac{1}{4}+x-x^{2}$. For any real number $x$, define a sequence $(x_{n})$ by $x_{0} = x$ and $x_{n+1} = f(x_{n})$. If the sequence converges, let $x_{\infty }$ denote the limit.

The Fibonacci numbers $f_{1}, f_{2}, \ldots$ are defined recursively by $f_{1} = 1$, $f_{2} = 2$, and $f_{n+1} = f_{n}+f_{n-1}$ for $n \geq 2$. Show that

exists, and evaluate the limit.

Note: See also Problem 7.5.20.

Prove that

For $n$ a positive integer, let $H_{n}$ denote the $n^{\text{th}}$ partial sum of the harmonic series

Let $k > 1$ be an integer. Prove that

where $\log k$ is the natural logarithm of $k$, and $C$ is a constant.

Suppose $x_{1}, x_{2}, x_{3}, \ldots$ is a sequence of nonnegative real numbers satisfying

for all $n \geq 1$. Prove that $\lim_{n \rightarrow \infty } x_{n}$ exists.

Let $(a_{n})$ and $(\varepsilon_{n})$ be sequences of positive numbers. Assume that $\lim_{n \rightarrow \infty } \varepsilon_{n} = 0$ and that there is a number $k$ in $(0,1)$ such that $a_{n+1} \leq ka_{n}+\varepsilon_{n}$ for every $n$. Prove that $\lim_{n \rightarrow \infty } a_{n} = 0$.

Prove or disprove (by giving a counterexample), the following assertion: Every infinite sequence $x_{1}, x_{2}, \ldots$ of real numbers has either a nondecreasing subsequence or a nonincreasing subsequence.

Let $b_{1}, b_{2}, \ldots$ be positive real numbers with

Assume also that $b_{1} < b_{2} < b_{3} < \cdots$. Show that the set of quotients $(b_{m}/b_{n})_{1 \leq n < m}$ is dense in $(1,\infty )$.

Which of the following series converges?

Let $a_{1}, a_{2}, a_{3}, \ldots$ be positive numbers.

For each $(a,b,c) \in \mathbb {R}^{3}$, consider the series

Determine the values of $(a,b,c)$ for which the series

For which real numbers $x$ does the infinite series

converge?

For which values of the real number $a$ does the series

converge?

Let $A$ be the set of positive integers that do not contain the digit 9 in their decimal expansions. Prove that

that is, $A$ defines a convergent subseries of the harmonic series.

Let $a_{1}, a_{2}, \ldots$ be positive numbers such that

Prove that there are positive numbers $c_{1}, c_{2}, \ldots$ such that

Evaluate the limit

Outline a proof, starting from basic properties of the real numbers, of the following theorem: Let $f: [a,b] \rightarrow \mathbb {R}$ be a continuous function such that $f'(x) = 0$ for all $x \in (a,b)$. Then $f(b) = f(a)$.

Let $f(x) = x\log (1+x^{-1})$, $0 < x < \infty$.

Let $f(x)$, $0 \leq x < \infty$, be continuous, differentiable, with $f(0) = 0$, and that $f'(x)$ is an increasing function of $x$ for $x \geq 0$. Prove that

is an increasing function of $x$.

Let $y: \mathbb {R} \rightarrow \mathbb {R}$ be a $C^{\infty }$ function that satisfies the differential equation

for $x \in [0,L]$, where $L$ is a positive real number. Suppose that $y(0) = y(L) = 0$. Prove that $y \equiv 0$ on $[0,L]$.