Doubt about the boundedness of convergent sequences theorem.

Question

I was reading up on the Theorem of boundedness of convergent sequences here and had a doubt on the last part of the proof, namely

Therefore for all $n\in N, |a_n∣ < M $ [...].

Shouldn't it be $\forall n \in N, |a_n| \leq M$ ?

It's a stupid question but I found the same thing in Alcock's "How to think about Analysis", so I wanted to be sure about which conclusion is correct.

Thanks in advance.

Answer 1

Yes, you are right. Let $a_1=17$ and $a_n=1/n$ for $n \ge 2$. With the notation in the proof we have $L=0$ and $M=17$.