Definition of Order in real Analysis

Question

In Walter Rudin's Principle of Mathematical Analysis 3ed, what's the difference between $<$ sign, which is used to denote the order relation and $<$ sign, which is used to compare $x$ and $y$ in first property in the definition 1.5 (i.e order's definition) ? My main problem is that i think order is the definition of $<$ (i.e operator which is used to compare two numbers) but how can the same operator (or relation, I don't know) $<$ (bottom red circle in image i attached) is used to define the same thing (top red circle in the image i attached) Definition 1.5?. What is the definition of $<$ sign used to compare two real numbers?

Answer 1

In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".

There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".

The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".

Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.