Prove that all convergent sequences are bounded

[converting1]

was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8

I have two questions,

why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?

 

[LCKurtz]

was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8

I have two questions,

why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?

Because $|x_n|< L+1$ only holds for $n > N$. If some of the first $N$ terms are larger than all the rest, use the biggest one of them for the bound.