NO;

1-My claim (using Universal quantifier) is different from yours(using existensial quantifier)!

When you say: There exists a number k in Z such that:...", it means that "for

**SOME** (at least one) k in Z,we have ..."; But it doesn't clear our mean which is "for

**EVERY** k in Z we have..."

Indeed yours is true but inadequate and doesn't make the real sense!

2-You need to know that the expression "x=2k*pi+_ a for all k in Z" is

**ANOTHER FORM** of the first expression which is used in some books and articles; And since you don't know this, you insist on your own interpretation of that and think they are different!!

I thought you know this and then i used it for the proof!

NOTE: Existential quantification is distinct from Universal quantification ("for all"), which asserts that the property or relation holds for

*any* members of the domain.

See:

Existential quantification - Wikipedia, the free encyclopedia