Can I make a citation only in the beginning of a chapter?
First, I would modify your sentence as (Mathoverflow etc. goes into the bibliography):


  The following proof is based on the ideas given by Iosif Pinelis [1].


Then,


  Do I need to cite Pinelis' answer whenever I use a notation/an idea of his post?


For the notation, you can make an addition to the sentence above (but explain the notation in the appropriate place):


  The following proof is based on the ideas given by Iosif Pinelis [1]. The notation employed in this section shall follow closely that of the cited work (or some equivalent sentence).


Instead, when using an idea, yes, I would put again the reference "[1]".