Q. 87 - Calculus

Question

Let $\left\{a_k\right\}$ be a sequence such that $a_{2k}\to L$ and $a_{2k+1}\to L$. Prove that $\left\{a_k\right\}\to L$. (What you will have proven is that if the even terms and odd terms of a sequence both converge to L, then the sequence converges to L.) 

Step-by-Step Solution

Answer

Proved that $\left\{a_k\right\}\to L$

Step 1. Given information

Consider the sequences $\left\{a_{2k}\right\}$ and $\left\{a_{2k+1}\right\}$ converging to limit L

Step 2. Prove that a k → L

The sequence $\left\{a_{2k}\right\}$ converges to limit L

Therefore,for given $\epsilon >0$there exists a positive integer N  such that

$\left|a_{2k}-L\right|<\epsilon for k\ge N$

The sequence $\left\{a_{2k+1}\right\}$ converges to limit L.
Therefore, for given $\epsilon >0$ , there exists a positive integer M such that $\left|a_{2k+}-L\right|<\epsilon for k\ge M$

Choose a positive integer P such that $P=max\left\{N,M\right\}$.
Therefore, for given $\epsilon >0$, there exists a positive integer P such that $\left|a_k-L\right|<\epsilon$ for $k\ge P$
Thus, the sequence $\left\{a_k\right\}$ converges to the limit L.
Hence, is the even terms and odd terms of a sequence both converge to L, then the sequence converges to L.