I found this results while doing my favourite activity - skimming through old books at library :) The first one is a result (n-above-a) and what follows is a theorem called Pasting Theorem in literature. n above a Let $a$ be a fixed real number. Then - Proof: $$\text{Part I}$$ Suppose first that $0 < a < 1.$ It follows that ${1 \over a} > 1$ and so we may write $${1 \over a} = 1 + h \quad \text{where } h > 0$$

I recently came across this question given in some olympiad paper and found it quite accessible, so here is the proof in all its glory after a lemma that will be in use. By the way, all the theorem says is that as long as $p \geqslant 1$ we can approximate any real number to a rational number by some arbitary precision. Lemma: Consider $n+1$ positive real numbers $x_0, x_1, x_2, x_3, \ldots, x_n$ all being in $[0,1)$ then for any $i \neq j$ and $i > j$ it so happens that $$\left | x_j - x_i \right | < \frac{1}{n}$$ for $i,j \in \mathbb{Z}^+$

This is the final part of the graph theory notes. We take a look at Tournaments. Tournaments So far all the graphs encountered were undirected that is, none of the edges that stood for “relationship” between two vertex had any direction. If we attach some directional information to an edge, then the graph is called a directed graph. Let the directional information be represented by an arrow which points to a vertex from another vertex, the from side has the arrow and is called an arc in literature.

This is the eight part of the graph theory notes. We take a look at some theorems mostly related to Ramsey Numbers. The book followed for the other notes is : Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings. For this post, the readers are also encouraged to seek the proofs in other books such as the legendary Adrian Bondy & U.

This is the seventh part of the graph theory notes. We take a look at Planar Graphs. The book followed throughout is: Graph Theory with Applications by Bondy J.A., Murty U.S.R.. Please refer to the book for examples as well as visual graph drawings. Planar Graphs A graph is called Planar if it can be drawn in the plane so that its edges intersect only at their ends. The vertices and edges of the graph partition (no overlap) the plane into several regions, each such region is called a face, denoted by $F_i$ for some $i \in \mathbb{N}$.