Condition (C) of Palais-Smale

Question

In Klingenberg's Notes, he makes the following definition:

$\Lambda M$ will be said to satisfy the condition (C) of Palais-Smale if:

Given a sequence $\{c_m\}$ on $\Lambda M$ satisfying:

(i) The sequence $\{E(c_m)\}$ is bounded

(ii) The sequence $\{||\text{grad} ~E(c_m)||_1\}$ tends to zero

Then $\{c_m\}$ has limit points and any limit point is a critical point of $E$

He then proceeds to prove $\Lambda M$ satisfies the condition (C). (OBS: $\Lambda M$ is the free loop space of a compact riemannian manifold)

But I think something is wrong... is it really limit point that he wants?

I think he means that $c_m$ has some convergent subsequence... and every subsequence limit is a critical point. Is that it?

Answer 1

He means that $c_m$ has some convergent subsequence.