Sequences — Question 2

*Question 2 Let $a_n=(-1)^n\left(1+\dfrac1n\right)$ for $n\ge1$.

1. Show that $(a_n)$ is bounded.

2. Find the limits of the even and odd subsequences $(a_{2k})$ and $(a_{2k-1})$.

3. Use those limits to decide whether $(a_n)$ converges and to evaluate the claim, “Every bounded sequence converges.”

*Question 2 – Solution

For $n\ge1$, $1<1+1/n\le2$. Since $|(-1)^n|=1$, $$|a_n|=1+\frac1n\le2.$$ Thus $-2\le a_n\le2$, and the sequence is bounded.

For even indices, $$a_{2k}=1+\frac1{2k}\longrightarrow1.$$

For odd indices, $$a_{2k-1}=-1-\frac1{2k-1}\longrightarrow-1.$$

If $(a_n)$ converged to $L$, every subsequence would also converge to $L$. The even and odd subsequences instead approach $1$ and $-1$, respectively. Since these limits differ, $(a_n)$ diverges.

This bounded divergent sequence is a counterexample to the claim that every bounded sequence converges. Boundedness together with monotonicity, however, would guarantee convergence by the Monotone Convergence Theorem.