Formula involving double integral over a disc?

Problem:

Can anyone help me out with the following problem:
I am given a uniformly continuous function : $g:\mathbb{R}^{2}\rightarrow [0,\infty )$ such that the following condition is satisfied:
$$\sup_{r> 0}\iint_{{x^{2}+y^{2}\leq r^{2}}}g(x,y)dxdy< \infty$$
The question is to prove that:$$\lim_{| (x,y)| \to \infty}g(x,y)=0$$

I tried to use polar coordinates instead of rectangular ones, but it didn't work out. Any help?
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 First, note that the limit intuiteively means that if you go really far from the origin, g becomes almost zero. if you first suppose by contradiction the limit does not hold and $g(x,y)=g(x(r,\theta),y(r,\theta))$, then there must be some constant angle theta along which $g_{\theta}(x(r),y(r))$ has nonzero limit, right? If you try to use polar coordinates, you get $\iint r g(x(r,\theta),y(r,\theta)), \theta\in[0,2\pi], r\in[0,\infty)$ Because g is always positive and uniformly continous, I expect this is larger than the integral over the radiant with theta such that there is a nonzero limit (admittedly, this needs to worked out a bit), which exists as assumed by contradiction. So then you just need something like $\int_0^\infty f(x) dx < \infty, f(x)>0 \iff f(x)\rightarrow0$, which is easier than what you had first. Hope that helps a bit!

 Tags analysis, calculus