Topologically equivalent metrics? Ceiling function of metric $d$ I am asked if the following metrics are topologically equivalent or not.

$(X,d)$ is a metric space and $d$ is the metric. Define $\lceil{d} \rceil (x,y)$ := $\lceil{d(x,y)} \rceil$:$X \times X \rightarrow [0, \infty)$. Are $d$ and $\lceil{d} \rceil$ topologically equivalent?
  Give a proof if so, or provide a counter example.

My attempt was to say we have some sequence ${x_n}$ that converges to some $x$. Then, I managed to say that if $d(x_n,x) \rightarrow 0$ then $\lceil{d(x,y)} \rceil \rightarrow 0$ so I have that bit okay(If my tactics are okay).
I am having trouble with showing the reverse; $\lceil{d(x,y)} \rceil \rightarrow 0$ implies $d(x_n,x) \rightarrow 0$.
Well, because $\lceil{d(x,y)} \rceil =0$ simply means that $x \in (-1,0]$ yes?
So it does not necessarily mean it converges to $0$. 
So I thought this means that they are "inequivalent" i.e. a counterproof but I cannot be sure if this works or qualifies as a counterproof...I mean, while I cannot guarantee it is $0$, I also have no means to say that it can never be $0$.
I guess I am basically stuck.
What should I do? Can someone please help me out? Thank you so much for your help, very much appreciated!

Hint: Consider the interval $[0,1]$ with the usual metric. What is the topology under the ceiling metric? 
