Cauchy condition without absolute value for convergence of a sequence

Question

Let $(a_n)$ be a sequence of non-negative real numbers, not necessarily increasing. The Cauchy criterion states that $(a_n)$ converges if and only if $\forall \epsilon >0$, there exists an $N$ such that $m >n > N$ implies that $|a_m - a_n | < \epsilon$.

Suppose one can show that for all $\epsilon > 0$, there is an $N$ such that $m > n > N$ implies $a_m - a_n < \epsilon$ (i.e. without the absolute value). Is this knowledge sufficient to conclude that $(a_n)$ converges? If not, what is a counter-example?

Answer 1

It is true. We prove it by contradiction. Suppose that $(a_n)_{n=1}^\infty$ is a sequence of non-negative real numbers such that for any $\epsilon>0$, there exists $N$ such that for all $m>n>N$, $a_m-a_n<\epsilon$. Seeking a contradiction, assume the sequence does not converge. Then there exists a number $\delta>0$ and a subsequence $(b_n)_{n=1}^\infty$ of $(a_n)_{n=1}^\infty$ such that $|b_m-b_n|>\delta$ for all $m>n$. By passing to yet another subsequence, we may assume that $(b_n)_{n=1}^\infty$ is either increasing or decreasing. If it is decreasing, then $b_{n+1}+\delta<b_n$ for all $n$. This implies that $$b_1>b_2+\delta >b_3+2\delta >\ldots >b_{n+1}+n\delta.$$ For large enough $n$, this would imply that $b_{n+1}$ is negative. In particular, it is true for any $n$ so large that $n\delta>b_1$. Thus the sequence cannot be decreasing.

But the sequence also cannot be increasing. If it were, that would mean that $b_p>b_q+\delta$ for all $p>q$. But by the original hypotheses, there exists $N$ such that for all $m>n>N$, $a_m-a_n<\delta$. Then we can fix $p>q>N$ and let $m>n>N$ be such that $b_p=a_m$ and $b_q=a_n$ (we can do this, since $(b_i)_{i=1}^\infty$ is a subsequence of $(a_i)_{i=1}^\infty$). Then $b_p-b_q>\delta$ by the properties of $(b_i)_{i=1}^\infty$, but $b_p-b_q=a_m-a_n<\delta$ by the properties of $(a_i)_{i=1}^\infty$ and $N$.

Answer 2

A direct proof: For $\epsilon > 0$ choose an index $N$ such that $$ a_m < a_n + \epsilon \text{ for } m > n \ge N \, . $$ Taking $\limsup_{m\to \infty}$ implies that $$ \limsup_{m\to \infty} a_m \le a_n + \epsilon \text{ for } n \ge N \, , $$ and taking $\liminf_{n\to \infty}$ implies that $$ \limsup_{m\to \infty} a_m \le \liminf_{n\to \infty} a_n + \epsilon \, . $$ This holds for all $\epsilon > 0$ so that $\limsup a_n = \liminf a_n$, i.e. the sequence is convergent.

It suffices to require that the sequence is bounded below. Without that condition, one can still conclude that $(a_n)$ is convergent or diverges to $-\infty$.