Is the Limit of a Convergent Sequence in [0,1]?

[ptolema]

Homework Statement

suppose that {a_n} is a convergent sequence of points all in [0,1]. prove that lim a_n as n-->$$\infty$$ is also in [0,1]

Homework Equations

for all$$\epsilon$$>0, $$\exists$$ a natural number N such that for all natural numbers n, if n>N, then absolute value(a_n-L)<$$\epsilon$$

The Attempt at a Solution

i messed with the limit a lot, but the furthest i could get was that [0,1] was a subset of (-$$\epsilon$$,1+$$\epsilon$$), which contains L (the limit). can someone shed some light on this?

 

[HallsofIvy]

Suppose L were NOT in [0, 1]. Then either L< 0 or L> 1.

If L< 0, let $\epsilon= |L|/2$.

If L> 1, let $\epsilon= (L-1)/2$.