# subsequences

• Jan 13th 2010, 01:19 AM
alexandrabel90
subsequences
can someone explain this proof to me?

theorem:
let sn be a convergent seq with sn -> l as n tends to infintiy. then every sunseq pf sn converges to l.

from my notes, it states that by induction, you can easily establise that nk > k for all k....

sorry, im new to the whole idea of proofs so im not sure how do i use induction to prove that nk > k..

thanks
• Jan 14th 2010, 01:05 AM
Laurent
Quote:

Originally Posted by alexandrabel90
from my notes, it states that by induction, you can easily establise that nk > k for all k....

I guess you mean: if $\displaystyle (n_k)_{k\geq 0}$ is a strictly increasing integer-valued sequence, then $\displaystyle n_k\geq k$ for all $\displaystyle k\in\mathbb{N}$.

Since $\displaystyle n_0\in\mathbb{N}$, we have $\displaystyle n_0\geq 0$, this is the base case.
Let $\displaystyle k\in\mathbb{N}$. Assume that $\displaystyle n_k\geq k$. Let us prove that $\displaystyle n_{k+1}\geq k+1$. Because $\displaystyle (n_k)_k$ is strictly increasing, $\displaystyle n_{k+1}>n_k$, hence $\displaystyle n_{k+1}>n_k\geq k$, and $\displaystyle n_{k+1}\in\mathbb{N}$, thus $\displaystyle n_{k+1}\geq k+1$. This concludes the induction.