# Why can negating '$\Phi(G) \geq \Omega(\phi)$' be converted to '$\Phi(G) <3\phi$' in this proof instead of '$\Phi(G) < c \cdot \phi$ for all $c > 0$'?

If we want to negate the statement '$$\Phi(G)$$ is at least $$\Omega(\phi)$$', the result is usually '$$\Phi(G) < c \cdot \phi$$ for all $$c > 0$$'.

However, I found that a contrapositive only indicates that $$\Phi(G) < 3\phi$$. Would it make the proof lose generality?

Source: EECS 498 FA 21 Theorem 2.9, proof of "$$\Rightarrow$$".

($$\Phi(G)$$ is the conductance of a graph $$G$$)

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• the negation is: for all $c > 0$, we have $\Phi(G) < c\cdot \phi$ for infinitely many graphs. idk why you're saying "why is the negation expressed as", as if that's a universal phenomenon. Commented yesterday
• I'm sorry for the confusion. I made changes to the question to make it clearer. I'm curious about the technique used for this particular proof. Commented 20 hours ago