1. sequences

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

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

and the second Q

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

Thanks guys!

2. Originally Posted by sfspitfire23
I'm studying for a test and am looking at 2 questions which I'd like guidance with

Suppose the sequence $c_n$ converges to $C$. Define the sequence $d_n=\frac{c_n+c_{n+1}}{2}$ Does the sequence $\{d_n\}$ converge or diverge?
If $c_n$ converges to $c$ then every subsequence ( $c_{n+1}$ in particular) converges to $c$

[quote

and the second Q

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

Thanks guys![/QUOTE]

Since $A\ne 0\implies |A|>0\implies \frac{1}{2}|A|>0$ we may take $\varepsilon=\frac{1}{2}|A|$.....so

3. Ah, so $|a_n-A|<\frac{1}{2}|A|$ and the answer follows by adding $A$to $-\frac{1}{2}|A|$

4. Originally Posted by sfspitfire23
Ah, so $|a_n-A|<\frac{1}{2}|A|$ and the answer follows by adding $A$to $-\frac{1}{2}|A|$