Analysis of Universal Hashing
The statement of the corollary is extremely sloppy. Here is a correct statement and proof.


  Corollary 11.4
  
  Fix $c &lt; 1$. Using universal hashing and collision resolution by chaining in an initially empty table with $m$ slots, it takes expected time $\Theta(N)$ to handle any sequence of $N$ Insert, Search, and Delete operations containing at most $cm$ Insert operations.
  
  Proof Since the number of insertions is at most $cm$, we have $n \leq cm$, where $n$ is the number of elements in the table, and so the load factor satisfies $\alpha \leq c$. The Insert and Delete operations take constant time and, by Theorem 11.3, the expected time for each Search operation is $O(1)$. By linearity of expectation, therefore, the expected time for the entire sequence of  $N$ operations is $O(N)$. Since each operation takes $\Omega(1)$ time, the $\Theta(N)$ bound follows. $\quad\blacksquare$


Roughly speaking, there are two problems with the book version:


The variable $n$ is used both for the number of elements in the table and for the number of operations.
The number of Insert operations should be $O(m)$, with a hidden constant bounded away from $1$.