How to prove a number is an accumulation point of a set?

A number xx is said to be an accumulation point of a non-empty set A⊆RA⊆R if every neighborhood of xx contains at least one member of AA which is different from xx.

A neighborhood of xx is any open interval which contains xx.

In this question, we have A=QA=Q and we need to show if xx is any real number then xx is an accumulation point of QQ. This is almost obvious because if xx is any specific real number then any neighborhood BB of xx contains infinitely many rational numbers (and hence at least one of them is different from xx itself).

The fundamental property which we are using here is the following:

If a<ba<b are two real numbers then there is a rational xx with a<x<ba<x<b and an irrational number yy with a<y<ba<y<b.

This above fact implies that there are infinitely many rational and irrational numbers between aa and bb. In other words any interval (a,b)(a,b) contains infinitely many rational and irrational numbers. The neighborhood BB in my answer above is an interval of this type and hence contains many rational numbers.

points are handed out whenever you post answers.

brainliest answers are handed out by the person asking the question, so whether your answer is picked depends entirely on whether the asker bothers to pick

if you're asking about your rank, it depends on having a certain number of points and a certain number of brainliest answers. for example, at the time of this writing, i have 44338 points out of 15000 needed for the next (nearly triple the requirement! ) but i only have 24 out of 50 required brainliest answers. this number decays, meaning after a certain amount of time (i think about a month), the total number of brainliest answers you've earned doesn't count toward this total. (i've been stuck at the same rank for a long

the lesson to learn here is that you shouldn't worry about points or rank. chances are, no matter how good an answer you may provide, your efforts will go mostly unnoticed unless you focus on the newest/most active questions that get posted.

answer: there are x+184 pages in olivia's book.

step-by-step explanation:

let the total number of pages in the book which olive has already read be x.

then if olive has 184 pages left to read, then the total number of pages are in olive's book= pages she has read+pages she has left to read

therefore, the total number of pages in olivia's book=

hence, there are x+184 pages in olivia's book.