Triangle inequality involving distance between point and set

Question

I am trying to prove that if $(X,d)$ is a metric space and $A$ a non-empty subset of $X$, then $\forall x,y \in X$, it is satisfied that $d(x,A) \leq d(x,y)+d(y,A)$. Obviously, in the particular case of $x=y$ we reach the equality and it resembles the triangle inequality but, as the distance of a point and a set is defined like $d(x,A)=\inf\{d(x,a)\mid a \in A\}$, I don't know how to apply formally the triangle inequality. I have thought also of using properties of infima but I don't know which. Thanks.

Answer 1

Let $a \in A$. Then $$d(x,a) \leq d(x,y) + d(y,a).$$ Now take the infinimum over $a$ on both sides.