given x, y > 0

considering option 1:
take x = 3, y = 1 (these values satisfy x > 2y and y < 2)
take x = 10, y = 4 (these values satisfy x > 2y, but does not satisfy y < 2)
so, we can eliminate option A

considering option 2:
take x = 2, y = 1 (these values satisfy x < y + 2 and y < 2)
take x = 10, y = 12 (these values satisfy x < y + 2, but does not satisfy y < 2)
so, we can eliminate option B

taking 1 and 2:
x > 2y
y + 2 > x
implies, y + 2 > x > 2y
y + 2 > 2y
2 > y , which is what we are looking for.

hence, it is C