Thank you for your reply. I'll check the website you mentioned, ask any question that I have at that website.

Thank you for your reply. I agree with you that this estimate already applies to the norm. I also agree with you about not being able to use a reasoning based on Taylor series. I'm not a mathematician, I'm a mechanical engineer working on control theory. I saw this inequality mentioned as a lemma on a paper which I read, where it was just mentioned that its proof was quite obvious. But I was just wondering how to show it.