Finding supremum of a set; am I underdoing it? 
Find, with proof, the supremum of the subset $E$ of $\mathbb{R}$ defined by:
$E = \left\{\frac{25+16\sqrt{n}}{5+3\sqrt{n}}:n\in\mathbb{N}\right\}$

Is it ok to solve the limit of the function and call it a proof?
i.e. $$\text{sup}(E) = \lim_{n\to\infty} f(n) =\space ...$$
Or do I need to involve actual definitions of bounds etc.

You have an increasing sequence
$$ \left\{\frac{25+16\sqrt{n}}{5+3\sqrt{n}}:n\in\mathbb{N}\right\}$$
with the limit as $n$ goes to infinity equal to $16/3$ 
The limit is the supremum, that is $$\sup (E) = 16/3$$
All you have to do is to show that the sequence is increasing and bounded above by its limit.
