In the Wikipedia article for $CAT(k)$ spaces, the first example says:
Any $CAT(k)$ space is a $CAT(l)$ space for all $l>k$. In fact, the converse holds: if a space is a $CAT(l)$ space for all $l>k$, then it is a $CAT(k)$ space.
In which direction is this false? I presume this is false, since if it were true, then any $CAT(k)$ space is also a $CAT(l)$ space for every $l>k$ implies it is a $CAT(l)$ space for every $l>k-1$, so it must also be a $CAT(k-1)$ space. If this were the case, people would probably just call them $CAT$ spaces. If my presumption is incorrect and I am misinterpreting something, I would love to be corrected. Thank you in advance!
Can't $k$ be real? It's not merely integral, no? You can't induct down from $k$ to $k−1$: there's stuff in between.