Examples of Proving Limits of Sequences

Examples of Proving Limits of Sequences

We will now look at some examples of proving the limits of sequences. Recall that a sequence $( a_n )$ is convergent to the real number $L$ and we write $\lim_{n \to \infty} a_n = L$ if $\forall \epsilon > 0$ there exists a $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$. We will now apply this definition to formally prove the limits of sequences.

Example 1

Let $\left ( \frac{1}{n} \right)$ be a sequence. Prove that $\lim_{n \to \infty} \frac{1}{n} = 0$.

Let $\epsilon > 0$ be given. We want to find a natural number $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid = \biggr \rvert \frac{1}{n} \biggr \rvert < \epsilon$.

We note that if $n ≥ N$, then $\frac{1}{n} ≤ \frac{1}{N}$ and so if we choose $N$ such that:

(1)
\begin{align} \frac{1}{n} ≤ \frac{1}{N} ≤ \epsilon \end{align}

Then $\frac{1}{n} ≤ \epsilon$. By the Archimedean property we know that such an $N \in \mathbb{N}$ exists since $\epsilon > 0$. Therefore $\lim_{n \to \infty} \frac{1}{n} = 0$.