I have a trouble in the proof to $EVP$ theorem:
About the existence of the $\lim (\varphi(y_n))$ ?
Any hints would be appreciated.
2026-04-22 23:18:06.1776899886
A trouble about the Ekeland variational principle
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By construction, the sequence $\varphi(y_n) + 2^{1-n}$ is non-increasing. It is also bounded from below (since $\varphi$ is bounded from below by assumption). It follows that $\varphi(y_n) + 2^{1-n}$ converges to some value $c\in \mathbb R$. Now $2^{1-n}$ converges to zero, thus $$c = \lim_{n\to \infty} \left(\varphi(y_n)+ 2^{1-n}\right) = \lim_{n\to \infty} \varphi(y_n).$$