Continuous preferences

I’m going thorugh your slides and the notes for HW3, Q7. I don’t understand the notation for continuous preference relationships.

What are xsubn (in your slides) or xsupn (in the notes)? Do these subscripts/superscripts refer to the different dimensions x and y? Do they refer to different values which x and y may take on? Do they refer to something else entirely?

They refer to elements of a sequence of elements in X. That is, x_1 , x_2 , … are each elements of X, as are y_1 , y_2 , … .
Continuity obtains iff for all such sequences where
(1) x_i is preferred to y_i for all i,
(2) lim i -> infty x_i exists, and
(3) lim i -> infty y_i exists
Then lim i-> infty x_i is preferred to lim i-> infty y_i

OK, my intuition is that a lexicographic preference relationship is not continuous, but I don’t know how  I’d prove that.

Eg,

x_i = (.5 + 1/2i , .25)
Lim x_i = (.5, .25)

y_i = (.5 – 1/2i , .75)
Lim y_i = (.5, .75)

For all i, x_i preferred to y_i, but lim y preferred to lim x.

Sorry for terse, but sent from phone…

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