Intereting Posts

The Power of Taylor Series
How do you calculate that $\lim_{n \to \infty} \sum_{k=1}^{n} \frac {n}{n^2+k^2} = \frac{\pi}{4}$?
Does $\sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n}$ converge?
Truth of Godel's sentence in standard interpretation
Integrating $\frac{\log(1+x)}{1+x^2}$
List of interesting math podcasts?
Math Analysis Designing Algorithms
Problem similar to folland chapter 2 problem 51.
Does there exist a Noetherian domain (which is not a field ) whose field of fractions is ( isomorphic with ) $\mathbb C$ ?
What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?
The probability that a random (real) cubic has three real roots
Existence of smooth function $f(x)$ satisfying partial summation
Why can a matrix whose kth power is I be diagonalized?
Compute $\sum_{k=1}^{\infty}e^{-\pi k^2}\left(\pi k^2-\frac{1}{4}\right)$
Any Set of Ordinals is Well-Ordered

Assume $f$ is integrable over $[a,b]$ and $\epsilon > 0$. Show that there is a step function $g$ over $[a,b]$ for which $g(x) \leq f(x)$ for all $x \in [a,b]$ and $\displaystyle \int_{a}^b (f(x)-g(x))dx < \epsilon$.

I am having trouble coming up with a step function that satisfies the second condition. Given any $f(x)$, it is easy to come up with a step function such that $g(x) \leq f(x)$ for all $x \in [a,b]$. But how do we deal with the second condition?

- Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?
- Relationship Between Sine as a Series and Sine in Triangles
- Finding parametric curves on a sphere
- If $f(2x)=2f(x), \,f'(0)=0$ Then $f(x)=0$
- How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?
- The limit as $x \to \infty$ of $ \frac {\sqrt{x+ \sqrt{ x+\sqrt x}} }{\sqrt{x+1}}$

- Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $
- Deriving the addition formula of $\sin u$ from a total differential equation
- Evaluating this integral : $ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $
- Deriving the addition formula for the lemniscate functions from a total differential equation
- Why is a straight line the shortest distance between two points?
- Intermediate Value Theorem and Continuity of derivative.
- A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$
- Exact smoothness condition necessary for differentiation under integration sign to hold.
- The big $O$ versus little $o$ notation.
- A few questions on nonstandard analysis

This follows from the definition of Riemann integral:

For given $\epsilon>0$ there exists $\delta>0$ such that for every partition of $[a,b]$ that is finer than $\delta$, the lower and upper Riemann sum for that partition differ by less than $\epsilon$ from $\int_a^bf(x)\,\mathrm dx$, which is between them. Let $g$ be the step function corresponding to the lower Riemann sum. Then $g(x)\le f(x)$ for all $x$ and the lower Riemann sum is just $\int_a^bg(x)\,\mathrm dx$. Hence $\int_a^b(f(x)-g(x))\,\mathrm dx<\epsilon$, as desired.

- Approximating commuting matrices by commuting diagonalizable matrices
- What are the rings in which left and right zero divisors coincide called?
- Proving a particular subset of $R^n$ is closed
- Is there a name for these oscillations in the self-similarity of a set under the action of a cyclic group?
- Find number of integral solutions of $\sqrt{n}+\sqrt{n+7259}$
- Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?
- First order variation and total variation of a function/stochastic process
- Log integrals I
- General Triangle Inequality, distance from a point to a set
- Prove that there are infinitely many pairs such that $1+2+\cdots+k = (k+1)+(k+2)+\cdots+N$
- Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$
- Maximisation problem
- Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?
- Convergence in measure of products
- Finding cosets of $(\mathbb{Z}_2\times \mathbb{Z}_4)/\langle (1,2)\rangle$