Palais-Smale and Cerami

Question

I have this two definitions :

1. "$\phi$ satisfies the Palais-Smale compactness condition at the level c, or $(PS)_c$ for short, if every sequence $(u_j)\subset W$ such that $\phi(u_j)\rightarrow c$, $\phi'(u_j)\rightarrow0$ called a $(PS)_c$ sequence, has a convergent subsequence"

2)"$\phi$ satisfies the Cerami condition at the level c, or $(C)_c$ for short, if every sequence suchthat $\phi(u_j)\rightarrow c$ ,$(1+||u_j||)\phi'(u_j)\rightarrow 0$ called a $(C)_c$ sequence, has a convergent subsequence"

I don't see clearly if $1)\Rightarrow 2)$ or $2)\Rightarrow 1)$

Can someone help me please ?

Answer 1

$$\phi'(u_j) = \frac{(1+\|u_j\|)\phi'(u_j)}{1+\|u_j\|}.$$ The denominator cannot tend to zero, and if the numerator goes to zero, then $\phi'(u_j) \to \ldots$