Recent content by 3.1415926535

PROOF: http://math.stackexchange.com/questions/152354/proof-of-int-limits-af-int-limits-mathbbrf1-a-for-the-lebesgue-inte

\text{ Let }f:\mathbb{R}\to [0,\infty] \text{ be a measurable function and }A\subset \mathbb{R}Then, show that \begin{equation} \int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A \tag{1} \end{equation} \text{ where ${1}_A$ is the characteristic function of $A$ defined as } \begin{equation}...

Just happy to help

Is this the equation you want to solve: x3^{2x+1}=9x? If so, x3^{2x+1}=9x\Leftrightarrow x(3^{2x+1}-9)=0\Leftrightarrow x=0\text{ or } 3^{2x+1}=3^2\Leftrightarrow 2x+1=2\Leftrightarrow x=\frac{1}{2}

I ask for the Proof of the L'Hôpital Rule for the Indeterminate Form \frac{\infty}{\infty} utilising the Rule for the form \frac{0}{0} The Theorem: Let f,g:(a,b)\to \mathbb{R} be two differentiable functions such as that: \forall x\in(a,b)\ \ g(x)\neq 0\text{ and }g^{\prime}(x)\neq 0 and...

Even though I would like a more direct approach 2-5 will suffice. Suppose that I want to prove that a Cauchy sequence x_n converges How can I create a sequence of nested intervals whose lengths go to 0 when x_n is not necessarily monotonous?

If by property 1 you mean the least upper bound property the point here is not to use it! I want a proof 2-3 without using 1,3,4,5

The completeness properties are 1)The least upper bound property, 2)The Nested Intervals Theorem, 3)The Monotone Convergence Theorem, 4)The Bolzano Weierstrass, 5) The convergence of every Cauchy sequence. I can show 1→2 and 1→3→4→5→1 All I need to prove is 2→3 I therefore need the proof...

In short, in the traditional approach to calculus(with ε-δ definition of limits, continuity etc.) "dx" has no meaning and is used only because Leibniz used it. In nonstandard analysis, this "dx" is taken to be any infinitestimal and there are both positive and negative infinitestimals.

The book Foundations of Infinitesimal Calculus by H. Jerome Keisler is a great introduction to Non standard analysis. I am not sure however if you have the mathematical maturity to read it...

I am sorry, I messed up with latex. That was not my hypothesis...

I will prove the false statement, that n\geq a\Rightarrow n!\geq a^n, n\in \mathbb{N}-\left \{ 0 \right \} with induction For n=1 1\geq a\Rightarrow 1!\geq a^1\Rightarrow 1 \geq a which is true. Suppose that n\geq a\Rightarrow n!\geq a^n, n\in \mathbb{N}-\left \{ 0 \right \} Then...

Nevermind, I solved it. Here is the solution First of all, \left \| \mathbf{a} +k\mathbf{b}\right \|=1\Leftrightarrow (\mathbf{a} +k\mathbf{b})^{2}=1\Leftrightarrow \mathbf{a}^{2} +k^{2}\mathbf{b}^{2}+2k\left \langle {\mathbf{a} ,\mathbf{b} } \right\rangle =1\Leftrightarrow\left \langle...

Homework Statement Let there be two vectors \mathbf{OA},\mathbf{OB}\neq\mathbf{0}If \exists k\in \mathbb{R} such as that \left \| \mathbf{OA} +k\mathbf{OB}\right \|=1 show that Area(OACB)\leq\left \| \mathbf{OB} \right \| (OACB:parallelogram) Homework Equations None The Attempt...

A very informal (and possibly incorrect) proof I just thought of: df=df(x(t),y(t))=f(x(t+h),y(t+h))-f(x(t),y(t))=f(x(t+h),y(t+h))-f(x(t),y(t+h))+f(x(t),y(t+h))-f(x(t),y(t))=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy\Leftrightarrow \frac{df}{dt}=\frac{\partial f}{\partial...