Q:

Does modifying metric of a space change the limit of a sequence in it?

Im reading Rudin's Principles of Mathematical Analysis and get this idea. In my opinion, doing it will remove or add some limit points in the space because this modification may change the radius of their neighborhood. 
Im not sure whether it will exert effects on a limit of a convergent sequence. For example, {(1+1/n)|n∈$N$} in $R$ usual does converge to 1 and will it still converge to 1 when its metric is modified by d(x,y)=|1/x-1/y|? Or another question that we know (0,1) is open in $R$ usual, can you turn it to a closed set in $R$ just by modifying its metric?

A:

Consider the following metric: \[d(x,y) = |f(x) - f(y)|\]
where $f$ is the function that swaps the real numbers $7$ and $903$ and leaves every other real number untouched.
With this metric, the sequence $x_n = 7 + 1/n$ converges to $903$, so yes, different metrics can have completely different limits of sequences.

