The phrase 'at the discretion of' is grammatically correct. It means that a particular person has the authority to make a decision.
At your discretion.
its to her discretion
At your discretion.
we leave it in your discretion to
Neither is correct. With "the discretion of... " you want to use "at." Here are several ways to use this expression. (1) You may bring your notebook computer and use it at your own discretion. (2) We can participation in the dance class at the discretion of the professor.
The correct spelling is from discreet, spelled discretion (privacy, secrecy).
The correct spelling is discretion.
'In the hope that' is the correct phrase.
This is not a sentence it is a phrase and as a phrase it is correct.
no
no
The correct phrase is "sufficient proof".