How can we prove that the following theorem is valid for almost perfect secrecy?
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 $\varepsilon &gt; 0$.