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.

As other people have pointed out, the notation $a \leq b$ means "$a < b$ or $a = b$." 
For example, the statement $x \leq 5$ means "$x < 5$ or $x = 5$."
Now if a person knew for sure that, say, $x < 5$ or that, say, $x = 5$, they would probably say this, rather than saying, less informatively, that $x \leq 5$. The usefulness of a statement such as "$x \leq 5$ comes from the fact that the person may not know which of these statements is true.
Since the statement 
$$x \leq 5$$ 
means "$x < 5$ or $x = 5$", the statement is certainly true when $x = 5$. Therefore the statement above must also be true when we replace $x$ with $5$:
$$5 \leq 5.$$
The reason that this statement, though indisputably true, now appears so odd is that, now that we know what $x$ is, there is no longer any point in hedging our bets. We might as well just say that $5 = 5$. 
The difficulty is not related to the fact that the statement "$5 < 5$ or $5 = 5$" is hard to interpret logically. It is that on a pragmatic level, we can no longer understand why a person would not just come out and say which of the two possibilities is the correct one.
But if mathematics can allow a statement such as "$x \leq 5$," it must also allow the statement "$5 \leq 5$," in which $x$ has been replaced by a number.
