$(15$ points$)$ Big$-$O$,$ Big$-,$ and Big$-,$ formal$.[$We are expecting: For each part, a rigorous $($but short$)$ proof, using the definition of Big$-$O$,$ Big$-,$ and Big$-$O$.]$ a$.loglog(n)=O(lo{g}^2(n))$ b$.n^3=(n^{2}logn)$ c$.n^2+5n\sqrt[\phantom{2}]{n}$ is not$(n^3)$ d$.5n^2+3nlogn+2n+5=O(n^2)$ e$.sin(n)=O(1)$