Statement that $a \geqslant a$ Is it legitimate to make a statement that $a \geqslant a$? This sign means greater or equal, and  surely the second part (equal) will always hold. But maybe someone will disagree and say that I must have used $a = a$ instead.

Assuming that the "context" of your question is mathematical logic, we must start with a language.
First-order number theory can be the appropriate one. This language has five primitive symbols : $0$, $S$ (the successor function), $+$ and $.$ (the sum and product functions) and the $=$ (equal) predicate, plus countable many individual variables ($v_i$).
We can assume also first-order Peano's Axioms.
In this language, the following are well-formed formulae :

$v_1 = v_2$, $\lnot v_1 = v_2$, $0=0 \lor \lnot 0=1$, $v_1 + v_2 = v_3$, and so on.

In this language, $>$ is not a primitive symbol; we must introduce it as an abbreviation :

$x > y$ stand for : $\exists z (x = y+S(z))$.

Now, we can introduce the expression $x \ge y$ as an abbreviation for : $x > y \lor x = y$.
The interesting thing to do is to prove (starting from Peano's Axioms) the following theorem of f-o number theory :

$\vdash \forall x (x \ge x)$.

