Every interval is closed. (I think my proof is not concise) Definition of limit point. A point $p$ is said to be a limit point of a point set $M$ if and only if every region containing $p$ contains a point of $M$ distinct from $p$. The set of all limit points of a point set $M$ is denoted $M'$.
Definition of closed set. A point set $M$ is closed means that if $p$ is a limit point of $M$, then $p$ belongs to $M$.
Definition of interval. If $x \prec y$, then the set consisting of $x$ and $y$ together with all the points between $x$ and $y$ is called the interval $xy$.
Every interval is closed.
Proof: Let $I$ be a closed interval. By definition, a set is closed if $p$ is a limit point of $M$, then $p$ belongs to $M$. We want to show that every interval is closed is true for $I$. 
Let $I=[a,b]$. Any arbitrary point in $I$ will belong to the set and so will each of it's limit points. Consider $a$, where any region around $a$ always contains a point distinct from $a$ in $I$, so $a$ is a limit point. The same goes for $b$ and all other points in the set and in between the set points. A region around the points can always contain a point distinct from the others. So all points of $I$ are limit points.
Therefore we know that every closed interval is a closed set.
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 A: Your definition of interval is unusual. By your definition, only closed intervals are intervals. This is different from the usual definition of interval, which permits intervals of the form $(a,b)$, for example. 
Using your non-standard nomenclature, the statement is correct, but the proof is circular. You say in your proof:  "A region around the points can always contain a point distinct from the others." This is exactly what are you trying to prove, so you can't assume it in the proof. You need to prove this using the definition of interval.
A: First, your definition of "interval" is for what I would call a closed interval and later you use the phrase "closed interval".  You say " Any arbitrary point in I will belong to the set and so will each of it's limit points"  The first part, that any point in I is in I is a tautology and the second part, that its limit points are in the set, is what you want to prove!  You say "consider a" and assert that "any region around a always contains a point distinct from a in I".  That's the crucial point! Why is that true?
