Sequences — Question 2

Question 2 – Solution

Step 1: Prove boundedness.

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.

Step 2: Analyze the even subsequence.

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

Step 3: Analyze the odd subsequence.

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

Step 4: Apply the subsequence criterion.

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.

Step 5: Evaluate the student’s claim.

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.