Hello everyone i'm trying to solve an exercise that contains the following istructions.
Let it be $ \Bbb R^n = \{ x = (x_1, ..., x_n) : x_1, ..., x_n \in \Bbb R \}.$
Let define on $ \Bbb R^n$ a relation $<$ as below:
for $x,y\in \Bbb R^n$ and $ x<y$ if, and only if exists a j $\in \{1,..,n \}$ with $(x_j<y_j)$
and for all i $ \in \Bbb N$ with $(i<j \to x_i=y_i).$
And what do the above istructions mean?
This is the lexicographical order.
This ordering is the same as the one that orders words in a dictionary. What makes a word come before another (of the same length) in the dictionary? Well that happens exactly when at the first place the two words differ, the word that comes first has a letter that comes before the other word's letter in the underlying alphabet.
Same thing here: we start with an order on $\mathbb{R}$, our 'alphabet', and then declare that a sequence, or 'word', of $n$ reals, or 'letters', comes before another if at the first place where the sequences differ, the first word's letter comes before the second's in the alphabet.
So for example when $n=3$, $(1,1,1)<(1,1,2)<(1,1,2.1)<(1,2,-5)<(1,2,1000)<(700,3,-13)$ etc., and when $n=4$, $(-3,4,4,1)<(-3,4,5,-6)<(-3,4,5,-5)$.