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- real analysis - Understanding the definition of Cauchy sequence . . .
My question is related with the definition of Cauchy sequence As we know that a sequence $(x_n)$ of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer
- Proving that a sequence $|a_n|\\leq 1 n$ is Cauchy.
You know that every convergent sequence is a Cauchy sequence (it is immediate regarding to the definitions of both a Cauchy sequence and a convergent sequence) Your demonstration of the convergence of the sequence is right, so yes, you have proven that $ (\vert a_n\vert)$ is a Cauchy sequence
- Understanding proof that $\mathbb {R}$ is Cauchy complete
4 Context: The real numbers were constructed using Cauchy sequences of rational numbers, where every real corresponds to the equivalence class of a rational Cauchy sequence The fact that $\mathbb {R}$ is Cauchy complete will be used later to prove the Least Upper Bound Property, so we can't use that here
- I would like to know an intuitive way to understand a Cauchy sequence . . .
Since you asked specifically how to understand Cauchy sequences "intuitively" (rather than how to do $\epsilon,\delta$ proofs with them), I would say that the best way to understand them is as Cauchy himself might have understood them
- linear algebra - Why does the Cauchy-Schwarz inequality hold in any . . .
Here is an alternative perspective: Cauchy-Schwarz inequality holds in every inner product space because it holds in $\mathbb C^2$ On p 34 of Lectures on Linear Algebra, Gelfand wrote: Any 'geometric' assertions pertaining to two or three vectors is true if it is true in elementary geometry of three-space Indeed, the vectors in question span a subspace of dimension at most three This
- linear algebra - Intuition for the Cauchy-Schwarz inequality . . .
Cauchy-Schwarz inequality in this case is just a simple consequence of solving the least square problem $\min_ {t\in\mathbb R}f (t)$ This is not "visual", but arguably very intuitive and elegant
- probability - How to prove the ‘covariance inequality’ for discrete . . .
However, they all either seem to be for continuous random variables, or just refer me to the Cauchy-Schwarz inequality, which I am aware of, but not sure how to apply to this particular proof Basically, I am wondering if there is a way to prove this inequality using those above definitions
- Cauchy-Schwarz inequality with Expectations - Mathematics Stack Exchange
Cauchy-Schwarz inequality has been applied to various subjects such as probability theory I wonder how to prove the following version of the Cauchy-Schwarz inequality for random variables: $$\\
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