Convergent Real Sequence/Examples/1 plus Reciprocal of n

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Example of Convergent Real Sequence

The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:

$a_n := 1 + \dfrac 1 n$

is convergent to the limit $1$ as $n \to \infty$.


Proof

Let $\epsilon \in \R_{>0}$ be given.

The requirement is to find a value of $N \in \R$ such that:

$\forall n > N: \size {\paren {1 + \dfrac 1 n} - 1} < \epsilon$

But we have:

\(\displaystyle \size {\paren {1 + \dfrac 1 n} - 1}\) \(=\) \(\displaystyle \size {\dfrac 1 n}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 n\) as $n > 0$

Hence the requirement is now to find a value of $N \in \R$ such that:

$\forall n > N: \dfrac 1 n < \epsilon$

From Reciprocal Function is Strictly Decreasing:

$\dfrac 1 n < \epsilon \implies n > \dfrac 1 \epsilon$

So choosing $N = \dfrac 1 \epsilon$ we have that:

$\forall n > N: n > \dfrac 1 \epsilon$

and so:

$\forall n > \dfrac 1 \epsilon: \size {\paren {1 + \dfrac 1 n} - 1} < \epsilon$

Hence the result.

$\blacksquare$


Sources