title: Prove that f(n) = Θ(g(n)) iff g(n) = Θ(f(n))
body: <p>I have been given the problem: </p>
  
  <pre><code>f(n) are asymptotically positive functions. Prove f(n) = Θ(g(n)) iff g(n) = Θ(f(n)). 
  </code></pre>
  
  <p>Everything I have found points to this statement being invalid. For example an answer I've come across states:</p>
  
  <pre><code>f(n) = O(g(n)) implies g(n) = O(f(n))
  f(n) = O(g(n)) means g(n) grows faster than f(n). It cannot imply that f(n) grows
  faster than g(n). Hence not true.
  </code></pre>
  
  <p>Another states:</p>
  
  <pre><code> If f(n) = O(g(n)) then O(f(n)). This is false. If f(n) = 1 and g(n) = n 
   for all natural numbers n, then f(n) &lt;= g(n) for all natural numbers n, so
   f(n) = O(g(n)). However, suppose g(n) = O(f(n)). Then there are natural
   numbers n0 and a constant c &gt; 0 such that n=g(n) &lt;= cf(n) = c for all n &gt;= 
   n0 which is impossible.
  </code></pre>
  
  <p>I understand that there are slight differences between my exact question and the examples I have found, but I've only been able to come up with solutions that do not prove it. I am correct in thinking that it is not able to be proved or am I looking over some detail? </p>