How can we prove that the following theorem is valid for almost perfect secrecy? We have the following theorem:  


  Let $\Pi$ be a perfectly-secret scheme over message space $M$, and let $K$ be determined by $Gen$. Then $|K| ≥ |M|$.


How can we prove that the above theorem is valid for almost perfect secrecy? The definition for almost perfect secrecy is as follows:


  The encryption scheme $\Pi = (Gen,Enc,Dec)$ over a message space $M$
  is almost perfectly secret or $\varepsilon$-perfectly secret
  if for every probability distribution over $M$, $\forall m \in M$ and
  $\forall c \in C$ for which $Pr[C = c] &gt; 0$ and for a constant
  $\varepsilon &lt; 1$:
  
  $$|Pr[M = m|C = c] - Pr[M = m]| &lt; \varepsilon$$


EDIT
You cannot prove that because it is not true. 

Chapter two of the book Introduction to Modern Cryptography explains about perfect secrecy and defines this $\varepsilon$-perfectly secret notion. In exercise 2.12 you are asked exactly to 


  [...] Show that $\varepsilon$-perfect secrecy can be achieved with $|K| &lt; |M|$
  when $\varepsilon &gt; 0$.