I'm studying for a test and am looking at 2 questions which I'd like guidance with

Suppose the sequence $\displaystyle c_n$ converges to $\displaystyle C$. Define the sequence $\displaystyle d_n=\frac{c_n+c_{n+1}}{2}$ Does the sequence $\displaystyle \{d_n\}$ converge or diverge?

and the second Q

Prove if $\displaystyle \{a_n\}$ converges to a nonzero real number $\displaystyle A$, then there exists $\displaystyle N$ such that $\displaystyle |a_n|\geq \frac{1}{2}|A|$ for all $\displaystyle n>N$

Thanks guys!