(adsbygoogle = window.adsbygoogle || []).push({}); 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 a_{i}(sorry about the underline, i dont know how to make bar over letters)

need to prove that for every a_{i}there exist b_{i}and n such that a_{i}b_{i}= 2^{n}

or a_{i}x10^{k}+ b_{i}= 2^{n}

we need 0 < b_{i}< 10^{k}

the problem becomes, prove that for every a_{i}, there exist n and k such that

a_{i}x10^{k}< 2^{n}< (a_{i}+1)x10^{k}

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 [itex]f(x): \mathbb{Q} \to \mathbb{Q} [/itex] there exist three bijective functions

[itex]\phi_1: \mathbb{Q} \to \mathbb{Q} [/itex] ,[itex]\phi_2: \mathbb{Q} \to \mathbb{Q} [/itex] and [itex]\phi_3: \mathbb{Q} \to \mathbb{Q} [/itex] satisfying [itex]f(x) = \phi_1 +\phi_2 + \phi_3 [/itex]

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 ?

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# Homework Help: Problems about numbers and bijective functions

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