Menger's Theorem is defined in Introduction to Graph Theory as follows:
Menger's Theorem. Let $u$ and $v$ be nonadjacent vertices in a graph $G$. The minimum number of vertices in a $u-v$ separating set equals the maximum number of internally disjoint $u-v$ paths in $G$.
Following the statement of the theorem, the authors give a really long inductive and by-cases proof. I find this puzzling because it seems to me that the following simple proof by contradiction would suffice:
Proof. Let $k$ be the cardinality of a minimum $u-v$ separating set, and let $l$ be the number of internally disjoint $u-v$ paths. Assume, to the contrary, that $l \neq k$. Then either $l < k$ or $l > k$. If $l < k$, then removing $l$ vertices from $G$, one vertex from each of the $l$ $u-v$ paths, disconnects $u$ and $v$ which is a contradiction. Similarly, if $l > k$, removing $k$ vertices will not disconnect $u$ and $v$; rather we would need to remove $l$ vertices to disconnect $u$ and $v$ which is, again, a contradiction.
Is there something I am missing?