Limit of sums is sum of limits in a metric space So I'm aware that in a normed space, the limit of the sums is the sum of the limits:

For normed space $(X, ||.||)$, if $x_n \rightarrow a$ and $y_n \rightarrow b$, then $(x_n + y_n) \rightarrow (a+b)$

But is this true in a general metric space $(X,d)$? The reason I can prove the above statement is that $||x_n + y_n|| - ||a+b|| \leq ||x_n + y_n - a - b|| < 2\epsilon.$
But is it similarly true in metric spaces that $d(x_n + y_n, a+b) \leq d(x_n,a) + d(x_n,b)$?
Is that clear? Thanks a bunch!
 A: 
But is it similarly true in metric spaces that  $d(x_n + y_n, a+b) \leq d(x_n,a) + d(y_n,b)$?

Even assuming that you have a notion of addition,  the above inequality need not hold. For example, define a metric on $\mathbb R$ by $d(x,y)=|10^x-10^y|$. Using the standard notion of addition, we have 
$$d(1+1,0+0) = 100> 20 = d(1,0)+d(1,0)$$ 
I'll try to explain.


*

*Look at the   definition of a norm: you will find that in involves addition of vectors. The definition requires the norm to "play well" with addition. 

*Now look at the definition of a metric. There is no mention of addition of elements of $X$, or any arithmetic with them. If you just introduce addition or other binary operation (or any other structure) on a metric space, you cannot expect the metric to   play well with them; the metric does not know anything about those structures. 


If you want two unrelated concepts to play well with each other, you have to define them so that they do. This is why the definition of a topological metric space requires addition to be a continuous operation. Without this requirement, the additive structure would have nothing to do with topology, making the definition rather useless.
