I'm having trouble looking at two problems dealing with sequence limits in advanced calculus. here they are:

A: Suppose that the sequence $\displaystyle a_n$ converges to a nonzero constant $\displaystyle A$. Prove that if $\displaystyle n \geq n*$, we have $\displaystyle |a_n| \geq t|A|$, where t is a fixed real number satisfying $\displaystyle 0 < t < 1 $. Is this statement true if $\displaystyle t = 0$? How about $\displaystyle t = 1 $ Explain.

I have looked at if $\displaystyle t=0$, it is not true because $\displaystyle 0 |A| \not= 0$, and if $\displaystyle t=1$, then it is true, $\displaystyle 1 |A| \geq 0 $. writing the proof has me confused.

B: Prove that if a sequence $\displaystyle a_n$ converges to 0, and a sequence $\displaystyle b_n$ is bounded, then the sequence $\displaystyle a_n b_n$ converges to 0.

I have looked at using the theorem that states if $\displaystyle \lim _{x\to \infty }a_n b_n = A B$, but it requires that $\displaystyle b_n $converges to $\displaystyle B$, what other methods should I use?