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Problem: I am stuck on the following problem and I appreciate if someone is willing to help solving it. The problem is as follows: I am given a uniformly continuous function : $f:\mathbb{R}^2 \rightarrow [0,\infty )$ such that the following condition is satisfied: $$\iint_{ R^2} f(x,y)\,dx\,dy< \infty .$$ The question is to prove that:$$\lim_{| (x,y)| \to \infty}f(x,y)=0$$ |
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Hint: Try a contrapositive proof. If the limit were not true, you'd have an $\epsilon > 0$ and $(x_n,y_n)$ of magnitude at least $n$ such that $f(x_n,y_n) > \epsilon$. Use the uniform continuity to show there's a little disk $B_n$ centered at $(x_n,y_n)$ such that the integral of $f$ over the disk $B_n$ is at least $\epsilon '$ for some fixed $\epsilon '$.... |
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