Hello all,

I have the following problem: suppose that $\displaystyle \{S_{n} \}$ is a convergent sequence. Then we are asked to prove that $\displaystyle \lim_{n\rightarrow \infty}2S_{n}$ exists.

I have done the following:

By assumption I know that $\displaystyle \{S_{n} \}$ converges. This means that $\displaystyle \forall \epsilon>0$ we can find an $\displaystyle N\in \mathbb{N}$ such that $\displaystyle \mid S_{n}-L \mid<\epsilon$ whenever $\displaystyle n\geq N$.

We therefore have:

$\displaystyle \mid S_{n}-L \mid<\epsilon$

$\displaystyle 2\cdot\mid S_{n}-L \mid<2\epsilon$

$\displaystyle \mid 2S_{n}-2L \mid<2\epsilon$

$\displaystyle \mid 2S_{n}-2L \mid-\epsilon<2\epsilon-\epsilon$

$\displaystyle \mid 2S_{n}-2L \mid-\epsilon<\epsilon$

Could someone point in the right direction?