How do I show the convergence of this integral? Does $I$ converge if $f(x,y)$ is bounded and continuous on $\mathbb{R}^2$?
\begin{align}
  I=\iint_{\mathbb{R}^2} f(x,y)e^{-(x^2+y^2)}dxdy
\end{align}
I tried to evaluate this integral value from the boundedness of $f$ and show it using squeeze theorem, but it did not work. How can I make it work?

Suppose $f(x, y)$ is bounded by $M$ i.e. $|f(x, y)| < M$.
$$
\begin{align}
\left|\iint_{\mathbb{R}^2} f(x, y) e^{-(x^2 + y^2)} dx dy\right| &\le \iint_{\mathbb{R}^2}\left| f(x, y) \right| e^{-(x^2 + y^2)} dx dy \\
&\le \iint_{\mathbb{R}^2} M e^{-(x^2 + y^2)} dx dy \\
&= M\int_\mathbb{R} e^{-x^2}dx \int_{\mathbb{R}} e^{-y^2} dy \\
&= M \pi
\end{align}
$$
which implies that the integral converges.
