1. ## sequences

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!

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 $\displaystyle c_n$ 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?
If $\displaystyle c_n$ converges to $\displaystyle c$ then every subsequence ($\displaystyle c_{n+1}$ in particular) converges to $\displaystyle c$

[quote

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![/QUOTE]

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

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

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