Is "proof by counterexample" a legitimate proof?

Question

Suppose that I am proving the following proposition:

Proposition 1. For some $a$, $f(x)=ax^2$ is not an increasing function.

I "proved" the proposition by providing a numerical counterexample.

My question is:

1. Can "proposition 1" be called as a proposition?

2. If I write a formal proof, starting with proof and end with Q.E.D., but everything in the proof is a counterexample, shall I call this "proof by counterexample" or "proof by contradiction"?

I am asking because I don't know if I should start the proof by saying "by contradiction" or "by counterexample".

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Motivation: From what I learnt we can prove things via counterexamples. But this wiki link says that proof by counterexample is not technically a proof. I think "proof by counterexample" is not a proof technique but counterexample is one thing we could use in a "proof by contradiction".

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Updated. I am mostly interested in the formal language and format of writing the proof, rather than the actual proof of the proposition 1.

Answer 1

A counterexample is (just as its name states) a particular case that shows that the theorem is invalid.

A contradiction shows that there is a logical inconsistency inherent in the proposed theorem. It need not include an example whatsoever.

Very different things.

Answer 2

Wikibooks are not necessarily up to the standard that, say, a Wikipedia article would be, and whatever this book is trying to say doesn't really make sense.

A counterexample to a statement is a complete and valid proof that the statement is false.

If I were to write up a proof of this statement, I would probably write something like the following:

Proposition. $f(x) = ax^2$ is not an increasing function for any value of $a$. (By "increasing" we mean strictly increasing, i.e. that $f(x) < f(y)$ for any $x < y$.)

Proof. Notice that, for instance, $f(-1) = a = f(1)$ for all $a$.