Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, ${a_n} = m$ Suppose that $(a_n)$ is a sequence of integers. Prove that $(a_n)$ converges if and only if
there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$,
${a_n} = m$. I thought this is the definition of convergence of a sequence. Anyone can guide me on how to prove this ?
Take any $0&lt;\epsilon&lt;\frac12$, and use the fact that convergent sequences are Cauchy to conclude that a convergent sequence of integers must be (eventually) constant. The other direction should be trivial.
That is wrong. Consider the sequence $a_n = 1 - 2^{n + 1}$, which converges to $1$. No integer $a_n$ in sight, no constant value for $a_n$ starting at some $N$ whatsoever.