Proof of the Cauchy-Schwarz Inequality: $|\mathbb{x} \cdot \mathbb{y}| \leq |\mathbb{x}| |\mathbb{y}|$

Proof

We will prove this by induction....

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A Proof of Bolzano–Weierstrass

Definition

For any infinite sequence $\{a_n\}, n \in \mathbb{N}$ that is bounded within the closed real interval $[x, y]$, there is a convergent subsequence....

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Question 10

Give an example of a family of intervals $A_n,n = 1,2,...$, such that $A_{n+1} \subset A_n$ for all $n$ and $\cap_{n=1}^\infty A_n$ consists of a single real number....

Did you know you can do mathematical induction over the reals? Don't listen to the naysayers. Follow along!...