Intereting Posts

Find the sum of $\sum 1/(k^2 – a^2)$ when $0<a<1$
Summable family in a normed linear space
Countable basis of function spaces
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
Geometric intuition of adjoint
Uniqueness in existence of a bilinear form
Is it possible to define an inner product such that an arbitrary operator is self adjoint?
Definitions of connected space
How to understand compactness?
Order of matrices in $GL_2(\mathbb{Z})$
convergence of sequence of averages the other way around
Ring isomorphism (polynomials in one variable)
Picard group of a smooth projective curve
Fibonacci using proof by induction: $\sum_{i=1}^{n-2}F_i=F_n-2$
Showing that the minimum distance between a closed and compact set is attained

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?

- Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$
- Behaviour of asymptotically equivalent functions after iterative exponentiation
- Show $\lim\limits_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log
- Why can't the second fundamental theorem of calculus be proved in just two lines?
- How to prove that $e = \lim_{n \to \infty} (\sqrt{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt{n\#} $?
- Closed form of arctanlog series

- How to evalutate this exponential integral
- Solving Trigonometric Derivatives
- Non-differentiability in $\mathbb R\setminus\mathbb Q$ of the modification of the Thomae's function
- Integral $I=\int \frac{dx}{(x^2+1)\sqrt{x^2-4}} $
- Find the value of $\lim_{x \to - \infty} \left( \sqrt{x^2 + 2x} - \sqrt{x^2 - 2x} \right)$
- Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$
- Why is gradient the direction of steepest ascent?
- Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent
- Stolz-Cesaro Theorem, 0/0 Case
- Moving a limit inside an infinite sum

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.

- $X$ is normal matrix and $AX=XB$ and $XA=BX$.why $A{X^*} = {X^*}B$ and ${X^*}A = B{X^*}$?
- Unions and Intersections of Open Sets are Open
- Matrix linear algebra generators
- $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
- Is there an isomorphism between factor groups and subgroups of symmetric groups?
- Composition of convex and power function
- Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
- Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous?
- 2D Integral of Bessel Function and Gaussians
- Finite rings without zero divisors are division rings.
- What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips?
- Find $\lim_\limits{x\to 0}{x\left}$. Am I correct?
- Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.
- Codimension 1 homology represented by Embedded Submanifold
- Evaluate $\int_0^\pi xf(\sin x)dx$