I'm trying to write the definition of functions only using logical symbols

Question

I'm trying to write the definitions of the following functions

1. ${n} \% {k} = {p}$ iff
2. $\lim_{n\to\infty} a_n = L$ iff
3. $(X, <<)$ is a poset. An element p is a maximal element of (X, <<) iff

What I tried,

For #1, since p is ther remainder of the division of n with k,

$$\forall{n}∈\mathbb{R}(\exists{k, p}∈\mathbb{R}(n | k = p))$$

For #2, The definition for a converging series is, $$\forall{\epsilon} > 0(\exists{N}∈\mathbb{N}(\forall{n}∈\mathbb{N}(n \geq N \Rightarrow |a_n - L| < \epsilon)))$$

For #3, The maximal element has to be the one on top $$\forall{n}∈ X(\exists{x}∈X(x \geq n))$$

I'm not sure if my definitions are correct