An application of mean value theorem in Banach spaces

Question

I am reading this paper and I did not understand a statement done in the final of the page $581$:

$$I_{Q_{\lambda}}(z_{\delta}) - I_{Q_{\lambda}}(u_n) \geq - \frac{1}{n} ||z_{\delta} - u_n||$$ and by mean value theorem, we then have $$\langle I_{Q_{\lambda}}'(u_n), z_{\delta} - u_n \rangle + o(||z_{\delta} - u_n||) \geq - \frac{1}{n} ||z_{\delta} - u_n||.$$

The mean value theorem gives

$$||I_{Q_{\lambda}}(z_{\delta}) - I_{Q_{\lambda}}(u_n)|| \leq ||I_{Q_{\lambda}}'(u_n)|| \ ||z_{\delta} - u_n|| + o(||z_{\delta} - u_n||).$$

How this last inequality give the penultimate inequality?

Thanks in advance!

Answer 1

I used Taylor's formula instead of the mean value theorem to obtain the inequality.

From Taylor's formula,

$$I_{Q_{\lambda}}(z_{\delta}) - I_{Q_{\lambda}}(u_n) = \langle I_{Q_{\lambda}}'(u_n), z_{\delta} - u_n \rangle + o(||z_{\delta} - u_n||).$$

Thus,

$$\langle I_{Q_{\lambda}}'(u_n), z_{\delta} - u_n \rangle + o(||z_{\delta} - u_n||) = I_{Q_{\lambda}}(z_{\delta}) - I_{Q_{\lambda}}(u_n) \geq - \frac{1}{n} ||z_{\delta} - u_n||.$$