Find the probability mass function of $Y=\min(X,c)$

Suppose that $X$ is a random variable with the geometric distribution. I want to find the probability mass function of $Y=\min(X,c)$ where $c$ is a real number.

I know that $X$ can be $0,1,...$ $$P(Y=y)=P(\min(X,c)=y)$$ $$f_X(x)=p(1-p)^{x}$$ And what happens when c is negative and $P(Y=c)$? I'm really confused and I don't know what to do. Would someone give me a hint?

$$\mathbb{P}(Y=y) = \begin{cases} p(1-p)^y, & \text{if $y\in\{0,1,2,\dots,c-1\}$ } \\ \sum_{y=c}^{\infty}p(1-p)^y, & \text{if $y=c$} \end{cases}$$

And what happens when c is negative ?

When $c\le 0$ you have that $Y$ is a degenerate rv in $c$ that is $\mathbb{P}[Y=c]=1$