Originally Posted by

**dwsmith** Use the def of convergence to prove the sequence converges.

Def: A sequence $\displaystyle \{a_n\}_{n\in\mathbb{N}}$ converges to an $\displaystyle A\in\mathbb{R}$ iif for each $\displaystyle \epsilon >0$ there exists $\displaystyle N\in\mathbb{N} \ \text{such that} \ \forall n\geq N \ \text{we have} \ |a_n-A|<\epsilon$

$\displaystyle \left\{5+\frac{1}{n}}\right\}_{n=1}^{\infty}$

So A = 5. Now, using the def.

Let $\displaystyle \epsilon >0$ and there is N such that for $\displaystyle n\geq N, \ \left | 5+\frac{1}{n}-5\right |=\left |\frac{1}{n}\right |<\epsilon$

$\displaystyle \Rightarrow \frac{1}{n}\leq\frac{1}{N}<\epsilon$

$\displaystyle \Rightarrow N>\frac{1}{\epsilon}$

Is that it?