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- Mar 10, 2012

- 835

1. $(x_n,y_n)\to (0,0)$.

2. $(x_n,y_n)\in P\cap Q$ for all $n$.

I conjecture that there is a continuous curve $\Gamma:[0,1)\to\mathbb R^2$ such that $\Gamma(0)=(0,0)$ and $\Gamma(t)\in P\cap Q$ for all $t>0$.

I am pretty sure that the above is true. But I have no ideas on how to go about proving it. If you draw a picture I think you will also be convinced that the above should be true. Can somebody help?