special relativity - Time dilation and Lorentz transformation

I have trouble understanding the Lorentz transformation to proof the dilation of time.

If I use:

$$dt ^{'} = \frac{dt}{\sqrt{(1-(v^2/c^2))}}$$

Understanding that $S^{'}$ is the moving frame of reference.

We can see that the square root in the denominator is smaller when $v \rightarrow c$ and we get that if $v \rightarrow c$ then $dt' \rightarrow \inf$... something different to the expected result that time goes slower in S'.

What am I doing wrong?

Many thanks in advance.