Notes - Analysis I MT22, Misc
Flashcards
What does it mean if $A$ and $B$ are both ordered fields and $A$ is dense in $B$?
For any $x, y \in B$, there exists $a \in A$ such that $x < a < y$.
What does a possible winding bijection from $\mathbb{Q} \to \mathbb{N}$ look like?
What’s a very quick, fun and bite-sized proof the harmonic series diverges?
Assume it has a limit $S$
\[\begin{aligned} S &= \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \ldots \\\\ &= \left(\frac{1}{1} + \frac{1}{2}\right) + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6}\right) + \left(\frac{1}{7} + \frac{1}{8}\right) + \ldots \\\\ &< \left(\frac{1}{2} + \frac{1}{2}\right) + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{6} + \frac{1}{6}\right) + \left(\frac{1}{8} + \frac{1}{8}\right) + \ldots \\\\ &= \left(\frac{1}{1}\right) + \left(\frac{1}{2}\right) + \left(\frac{1}{3}\right) + \left(\frac{1}{4}\right) + \ldots \\\\ &= S \end{aligned}\]then $S < S$, a contradiction.
What’s the $\varepsilon$-neighbourhood of a point $a$?
The interval $(a - \epsilon, a + \epsilon)$.
What does the Continuum hypothesis state?
There is no set $S$ such that
\[|\mathbb{N}| < |S| < |\mathbb{R}|\]Why is it very hard to prove the continuum hypothesis, that there is no set $S$ such that $ \vert \mathbb{N} \vert < \vert S \vert < \vert \mathbb{R} \vert $?
Because it’s unprovable in ZFC.