Showing Range of Sequence in Metric Space is Not Always Closed

[yifli]

Homework Statement

show that (the range of) a sequence of points in a metric space is in general not a closed set. Show that it may be a closed set.


2. The attempt at a solution
I don't know where to start.
For example, if we are given a sequence of real numbers and the distance between a and b is defined as |a-b|, it asks us to show that a sequence of real numbers is in general not a closed set?

 

[micromass]

Pick your favorite convergent sequence. Is this a closed set?

 

[HallsofIvy]

Note the difference between the sets {1, 1/2, 1/3,..., 1/n,...} and {0, 1, 1/2, 1/3, 1/4, ..., 1/n, ...}

 

[yifli]

Note the difference between the sets {1, 1/2, 1/3,..., 1/n,...} and {0, 1, 1/2, 1/3, 1/4, ..., 1/n, ...}

thank you. I can see {1, 1/2, 1/3,...} is not closed, but {0,1,1/2,1/3,...} is.