$y>3$ implies $y\geq 3$, or does it? 
$y\geq 3$, $y>3$

Implication and equality, is in the region of logic than mathematics.
If we take something easy like Germany and the EU:

Germany ⇒ EU

Because Germany is in the EU but the EU might be the UK or Sweden. (narrow goes to broad)
If we change it a bit:...

Germany ⇒ UK, Germany, Sweden...

It's STILL the same
Now let's change it to numbers

$2 \implies 2,3,4$

STILL the same, right?
So why does my mathematics teacher say this is false!:

$y>3 \to y\geq 3$

We can represent this as:

$4,5 \implies 3,4,5$

UPDATE:
For future users, my explanation is really bad and my knowledge limited. Look at the good answers instead.
 A: By definition, the statement “$y\ge 3$” is really shorthand for “$y>3$ or $y=3$.” If you assume that $y>3$ is true, then certainly “$y>3$ or $y=3$” is true, so $y\ge 3$ follows.
A: It may also help to prove to your teacher that $y \ge 3 \implies y > 3$ is not true for all $y$. To do this you simply need to come up with an example value for $y$ such that $y \ge 3$ is true while $y > 3$ is false. Clearly taking $y = 3$ will do.
[If your teacher is still not convinced, I would say he or she is incompetent to teach mathematics. You probably can't say that in public, so if it gets this far, politely ask your teacher to explain how his or her views align with the truth table for logical implication.] 
A: It is true that $y>3 \Rightarrow y\ge 3$ for all $y$, but most of everything else you have written fails to make sense.

Writing "Germany $\Rightarrow$ EU" or "Germany $\Rightarrow$ EU, Germany Sweden" makes no sense.
The symbol $\Rightarrow$ is used between propositions, claims that can be true or false. But "Germany" isn't a proposition. It makes no sense to ask whether Germany is true or false; nor does it makes sense to ask whether EU is true or false, or for that matter whether the word salad "EU, Germany, Sweden" is true or false. So these are not things that can be meaningfully written on the two sides of $\Rightarrow$.
Similarly neither "$2$" nor "$2,3,4$" nor "$4,5$" nor "$3,4,5$" is something that can be true or false, so these things cannot be written as arguments to $\Rightarrow$ either.
"$2\Rightarrow 2,3,4$" and "$4,5\Rightarrow 3,4,5$" are both nonsense, just like "Germany $\Rightarrow$ EU" is.
You can write $\{4,5\} \subseteq \{3,4,5\}$ and get a meaningful (and in fact true) statement out of it, but removing the set brackets and changing $\subseteq$ into $\Rightarrow$ does not result in a mathematically meaningful formula.
A: To see why $y>3\implies y\ge 3$, note that the interval $(3,\infty)$ is a subset of the interval $[3,\infty)$, just as Germany is a subset of the EU, so the statement "I am in Germany" implies the statement "I am in the EU".
To see that the implication does not work the other way around, note that $y=3$ satisfies $y\ge 3$, but not $y>3$.
A: It's certainly true that $y > 3 \Rightarrow y \geq 3$. However, the application of this can be subtle, which might be the source of the confusion between you and your teacher. For example: if the answer to a question is "$y > 3$", then "$y \geq 3$" will not be a correct answer - it would be like answering "In which country is Berlin?" with "Either Germany or Sweden". This is a technically true response, but it doesn't completely answer the question.
Likewise, if the correct answer to a question is "$y \geq 3$", then "$y > 3$" cannot also be correct - just because the right answer is a consequence of this answer doesn't mean that the right answer isn't a consequence of something else as well. To take an analogy, this is like answering "Which countries names begin with 'G'?" with just "Germany" - sure, everything on the list you gave is right, but not everything that's right is on there.
A: Aside from what @henning-makholm has answered. I would phrase your proof like the following.
Say $$y>3$$ then since $$y>3 \,\,\text{or}\,\, y = 3$$ is also a true statement (alas, $y>3$), we can say that $$y>3 \implies y\ge3$$

This is kind of similar to how $$y>3 \implies y>2$$ but do remember that just because the implication is true, it doesn't mean that $y>2$ is true.
A: The fact that $y>3 \implies  y\geq3$ is an example of the logical rule of inference known as disjunction introduction. For more info on this and other rules see here. Disjunction introduction states that for any statements $A,B$ (a statement is a true or false sentence), from $A$, we can deduce $A\lor B$. We write this as:
$$A \therefore A\lor B$$
(the symbol $\therefore$ is read as "therefore" or "implies"). 
Keeping in mind that $y \geq 3$ really means $y =3 \lor y>3$, you can substitute $(y>3) \equiv A, y=3 \equiv B$. Thus,
$$A\therefore (A \lor B)$$
$$(y>3) \therefore (y>3 \lor y=3)$$
The really cool thing about disjunction introduction is that it works for any two statements, $A,B$. For instance, we may use it to make the following deduction:
Socrates is a man.
Therefore, Socrates is a man or pigs are flying in formation over the English Channel. 
Disjunction introduction can be proved either via a truth table, or by the definition of the logical $\lor$ connective. Here is the truth table method:

