I would like to ask if there is any clear (in details) proof of the fact that if the dimension $ N \geq 2 $ and $ 1 < p < + \infty, $ then the space $ C_0^{\infty}( \mathbb{R}^N \setminus{ 0}) $ is dense in the Sobolev space $ W^{1,p}( \mathbb{R}^N). $ \
Remark: I found some answers to similar questions talking about truncature near zero and $ \infty, $ but I cannot see exactly the sequence that approximates any function $ u \in W^{1,p}( \mathbb{R}^N). $
Thanks in advance.
