Convergent Real Sequence/Examples/1 plus Reciprocal of n

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:

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.