Where did I make a mistake calculating the distribution of the sum of two independent exponentially distributed random variables?

Question

We know that if both $X$ and $Y$ are continuously distributed with PDFs $f^X$ and $f^Y$, then $X+Y$ is continuously distributed as well with density function $f^{X+Y}(x) = \int f^X(z)f^Y(x-z)\text{ d}\lambda(z)$ where $\lambda$ denotes the Lebesgue measure. Now, since $X,Y \sim \text{Exp}(\lambda)$, $f^{X+Y}(x) = \int_{0}^{\infty} \lambda e^{-\lambda z}\lambda e^{-\lambda (y-z)}\text{ d}z = \lambda^2\int_0^{\infty} e^{-\lambda z-\lambda y+\lambda z}\text{ d}z = \lambda^2\int_0^{\infty} e^{-\lambda y}\text{ d}z$ and this integral shouldn't exist. What did I do wrong here?