Subsequences that converge to different values

[R.P.F.]

Homework Statement

Is it possible to have a sequence that contains subsequences converge to every point in the infinite set {1, 1/2, 1/3,...}? If possible, then give an example; if not, give an argument.

Homework Equations

The Attempt at a Solution

I don't think it's possible. But I don't know how to argue that.
Any help is appreciated!

 

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There is already a thread on the same issue on the first page in this section.

https://www.physicsforums.com/showthread.php?t=432295