limits of the sum of two Gaussian distribution

Let $X_1$ and $X_2$ be independent random variables with distributions $\mathcal{N}(0,\sigma_1)$ and $\mathcal{N}(0,\sigma_2)$ respectively. Show that the distribution of $X_1 + X_2$ is $\mathcal{N}(0,\sigma_1 +\sigma_2)$.

In this question, I have tried to put convolution as both $X_1$ and $X_2$ are independent. Here applying the convolution formula as

$$f_Z(z) = \int_{-\infty}^{\infty} f_{X_1}(x_1) f_{X_2}(z - x_1) dx_1$$ But I am confusing the limits of the integration is it varies from $-\infty$ to $z$ or something else.

Another way to show that is to use Moment Generating Function and its properties.

$$M_{X_1}(t)=e^{\sigma_1^2 t^2/2}$$

$$M_{X_2}(t)=e^{\sigma_2^2 t^2/2}$$

and, by independence

$$M_{X_1+X_2}(t)=e^{\sigma_1^2 t^2/2}\cdot e^{\sigma_2^2 t^2/2}=e^{(\sigma_1^2+\sigma_2^2) t^2/2}$$

Which is the MGF of a $N(0;\sigma_1^2+\sigma_2^2)$