Answer and Proof for Supremum and Infimum questions

Question

I couldn’t find any answer on this 2 questions even I asked several mathematicians. Anybody here who can solve those?

https://i.stack.imgur.com/ehLv9.jpg

(Sorry for the link, I couldn’t upload the image from my phone)

Answer 1

I only speak English so I can only go by whats written in the set builder notation.

sup of the first set is obviously 2 because the maximum occurs when n = 1.

sup of the second set is 2 also. Graph the two functions f(x) = x+2 and g(x) = $x^{2}$. Find where g(x) begins to be greater than f(x). This is where they intersect. Which is at x = 2.

See if you can figure out the infimums on your own based on those. If you can't ill help.