grandes-ecoles 2013 Q3b

grandes-ecoles · France · x-ens-maths2__mp Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that for all $h, h' \in [0,1]$ such that $h \leq h'$, $\omega_{f}$ satisfies $$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by
$$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$
Show that for all $h, h' \in [0,1]$ such that $h \leq h'$, $\omega_{f}$ satisfies
$$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$

Paper Questions

Q1a Q1b Q1c Q2 Q3a Q3b Q3c Q4a Q4b Q5a Q5b Q5c Q5d Q6 Q7a Q7b Q8a Q8b Q9a Q9b Q9c Q10a Q10b Q11a Q11b Q12 Q13 Q14a Q14b Q15 Q16 Q17 Q18a Q18b Q19