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:
\(\ds \size {\paren {1 + \dfrac 1 n} - 1}\) | \(=\) | \(\ds \size {\dfrac 1 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 n\) | 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
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.5$: Example