Uniqueness of Limits of sequences I was reading about limits of sequences recently, and I came across this fact that, "there are situations in which sequences which may converge to more than one point.
Is that really possible ? Till now every real analysis book that i have referred to, has always stated that the limit of every sequence is unique.
I am just curious, if this is true then, doesn't this contradict the idea of uniqueness of limits of a sequence ?
What about limits of sequences in metric spaces, can this happen there too ?
 A: It is in fact true that limits of sequences aren't necessarily unique. Here's a simple example: Take $X := \{0,1\}$ with the pseudometric $d(x,y) = 0$ for all $x,y \in X$.
A sequence converges to a point $x$ if and only if it is eventually in every neighbourhood of $x$. But the only neighbourhoods in $X$ are the whole space, so every sequence in $X$ converges to both $0$ and $1$(!)
What went wrong? Because this clearly isn't how metric spaces behave, as you seem to have suspected. Metric spaces have the very nice property that any distinct pair of points has some positive distance $2\delta$ between them. This forces the open $\delta$-balls centered at each to be disjoint and thus a sequence converging to one cannot converge to the other.
In the more general setting of point set topology, we can use a similar argument to see that if a space is Hausdorff, then limits are unique.
A: In the trivial topology, every sequence converges to every point. The thing that is special about metric spaces is that they are Hausdorff (every point has distinct neighborhoods)
For a given set, one may put many different topologies, which is a collection of special sets satisfying certain properties, "on top" of a given space. In the metric space topology, we have that a set is open iff every point in it has a neighborhood - an "epsilon" from it and the boundary of the set (imagine something in $\mathbb{R}^2$). We could come up with funky topologies though, like the trivial topology, a collection of sets that satisfy our definition of a topology but capture less interesting information of our space (the convergence of a sequence for example tells us nothing).
Non Hausdorff spaces do exist though that matter to somebody, for example the Zariski topology: by defining the closed sets (complements of open sets) to be zeros of polynomials.  
A: The limit is unique if and only if you are in a Hausdorff space.
