Suppose \((b_n)_{n\ge1}\) is a sequence of positive real numbers with \[ b_n\to\infty, \qquad \frac{b_n}{n}\to0. \] Must there exist a sequence \((a_n)_{n\ge1}\) such that \[ \frac{a_1+\cdots+a_n}{n}\to0 \] and \[ \limsup_{n\to\infty}\frac{a_n}{b_n}=\infty? \] Give a proof or counterexample.
Solution
Yes.
For each \(n\ge1\), set \[ c_n=\sqrt{n b_n}. \]
1. One has\[ c_n\longrightarrow\infty. \]
Since \(b_n\to\infty\) and \(n\to\infty\), their product \(nb_n\to\infty\). Taking positive square roots gives the claim.
2. There exists a strictly increasing sequence of indices\[ n_1<n_2<\cdots \]such that for every \(k\ge2\),\[ c_{n_k}\ge\sum_{j=1}^{k-1}c_{n_j}. \]
1. Hence some \(n_k>n_{k-1}\) satisfies the required inequality.
3. Define\[ a_n= \begin{cases} c_{n_k},&n=n_k\text{ for some }k,\\ 0,&\text{otherwise}. \end{cases} \]Then, for every \(k\),\[ \sum_{j=1}^{k}c_{n_j}\le2c_{n_k}. \]
For \(k=1\) the inequality is immediate. If \(k\ge2\), step 2 gives \[ \sum_{j=1}^{k}c_{n_j} = \sum_{j=1}^{k-1}c_{n_j}+c_{n_k} \le 2c_{n_k}. \]
4. The Cesàro means of \((a_n)\) converge to zero:\[ \boxed{ \frac{a_1+\cdots+a_n}{n}\longrightarrow0 }. \]
If \(n_k\le n<n_{k+1}\), then by step 3, \[ \begin{aligned} 0 \le \frac{a_1+\cdots+a_n}{n} &= \frac{\sum_{j=1}^{k}c_{n_j}}{n} \\ &\le \frac{2c_{n_k}}{n_k} \\ &= 2\sqrt{\frac{b_{n_k}}{n_k}}. \end{aligned} \] Since \(b_n/n\to0\), the right-hand side tends to zero as \(k\to\infty\). Also \(k\to\infty\) whenever \(n\to\infty\), so the Cesàro means tend to zero.
5. Along the indices \(n_k\),\[ \frac{a_{n_k}}{b_{n_k}} = \sqrt{\frac{n_k}{b_{n_k}}} \longrightarrow\infty. \]
By definition, \[ a_{n_k}=c_{n_k}=\sqrt{n_kb_{n_k}}, \] which gives the displayed ratio. Since \[ \frac{b_{n_k}}{n_k}\longrightarrow0 \] and all terms are positive, its reciprocal square root tends to \(\infty\).
6. Therefore\[ \boxed{ \limsup_{n\to\infty}\frac{a_n}{b_n}=\infty }. \]
Step 5 shows that the ratio tends to infinity along the subsequence \(n=n_k\). Hence its limsup is infinite.
7. Q.E.D.
The sequence constructed in step 3 satisfies both required properties by steps 4 and 6.