Homework Help: Problems about numbers and bijective functions

1. Feb 1, 2012

Selfluminous

1. The problem statement, all variables and given/known data

1. Prove that any finite sequence of digits is a starting sequence of digits for some power of 2.

2. Relevant equations

3. The attempt at a solution
This one seems easy, i first thought about binary system ( sum of power of 2)
then: suppose the random finite sequence is ai (sorry about the underline, i dont know how to make bar over letters)
need to prove that for every ai there exist bi and n such that aibi = 2n
or aix10k + bi = 2n
we need 0 < bi < 10k
the problem becomes, prove that for every ai, there exist n and k such that
aix10k < 2n < (ai+1)x10k
i'm still working on it...
perhaps give me a hint on how to prove this or maybe there's a better way

1. The problem statement, all variables and given/known data
2. Prove that for any function $f(x): \mathbb{Q} \to \mathbb{Q}$ there exist three bijective functions
$\phi_1: \mathbb{Q} \to \mathbb{Q}$ ,$\phi_2: \mathbb{Q} \to \mathbb{Q}$ and $\phi_3: \mathbb{Q} \to \mathbb{Q}$ satisfying $f(x) = \phi_1 +\phi_2 + \phi_3$

2. Relevant equations

3. The attempt at a solution
I don't even know where or how to start with this one
should i use contradiction or something ?