That can not be rationally disputed. One need only consult hundreds if not thousands of math texts that describe the standard definition of limits.
I could see your saying that mathematicians have gotten it wrong. But I can’t see your saying that they didn’t say what they did say!
It is not necessary for a sequence or sum to “attain” its limit. That’s the whole point of limits. Attainment is NOT part of the definition. Arbitrary closeness is.
Likewise, SOME properties are preserved when we pass to the limit, and others aren’t. For example in the sequence 1/2, 1/3, 1/4, 1/5 1/6, … each term is strictly greater than 0, But the limit is 0. It is a fact taught to math majors that when we pass to the limit, (<) becomes (\leq) and (>) becomes (\geq).
The terms of 1/2, 1/3, 1/4, … get arbitrarily close and STAY arbitrarily close to 0. That is the definition of the limit. You have faulty ideas because you haven’t grokked the formal definition of a limit. That’s all I can see of your objection.
That’s just something you made up. It’s not mathematically true. Limits are based on arbitrary closeness, not attainment. It’s perfectly clear that 1/2, 1/3, … never “attains” the value 0. The limit is 0 because the terms get arbitrarily close to 0. That is the definition. And even though each element is strictly greater than 0, the limit is 0. That’s how limits work.