title: prove n = Big-O(1) using induction
body: <p>I know that the relation n = Big-O(1) is false. But if we use induction involving Big-O it can be proved. But the fallacy is we cannot induct Big-O. But my question is how we can disprove the relation by using the constants.</p>
  
  <p>The false proof is here, please give me the proof of it being false using the constants. I'm getting confused with the constants, I don't know whether each relation used in the proof is having different constant or the same. Please enlighten on the topic.</p>
  
  <pre><code>TO prove: n= O(1)
  for n=1 , 1= O(1) proved
  </code></pre>
  
  <p>induction hypothesis : let it is true : n-1 = O(1)
  now we prove that n = O(1)</p>
  
  <pre><code>LHS :  n = (n-1) + 1 
           =  O(1) + 1
           =  O(1) + O(1)
           =  O(1) 
  </code></pre>
  
  <p>Falsely proved..
  I want the clarification of the fallacy in terms of &lt;= and constants, that is in the basic definition of Big-O.</p>