@Koro If you come here please type @TeresaLisbon

@TeresaLisbon

Great.

I am trying to write more points as you suggested

@Koro Good. Don't go too far, just to get intuition.

to see how $\log_2n$ comes. By Induction, I have verified though that indeed $\alpha(n)\le \log_2n$

@Koro But you will need intuition as to why log_2 n comes ,right?

Exactly.

That's where I need your help. Thanks a lot for sparing time for helping me understand.

@Koro Sure, just give me a minute to type things out. Thanks

The answer to that is as follows : see , imagine a number $N$, and write it as its prime factors $N = p_1^(n_1) p_2^(n_2) ... p_k^(n_k)$ where the $p_i$ are distinct primes. What is $alpha(N)$? It is $k$ . Now, the point is that all the $p_i$ are at least $2$. In particular, $p_1>= 2, p_2 >= 2, ... , p_k >= 2$. So trivially, we have $N >= 2^(n_1) 2^(n_2)...2^(n_k) = 2^(n_1+n_2+...+n_k)$. Now take the logarithms and we get $log_2 N >= n_1+n_2+...+n_k$, but all the $n_k$ are at least $1$.

Let me process this

@Koro So $n_1+n_2+...+n_k >= 1+1+...+1 = k$. Putting it together, $log_2 N >= k = alpha(N)$.

The intuition here was that all the p_i are at least two, and once you make that reduction the exponents cannot be too large.

I understood the log part. Amazing

Now I'm writing a complete proof involving epsilon delta

Great. Do that, and call me there, I will up vote if your answer is good (at the moment I don't have any doubt about that, of course).

@Koro

Ok. I got it. Thanks a lot :)

I'll sure post my answer to my question.

Great, good talking to you

@Koro

I have one more doubt sir. It's related to an uncountable set say R and vacuous statement.

@Koro Go on.

@TeresaLisbon: I believe that the statement-"Two consecutive real nos. are transcendental" is true vacuously and so is the statement -" Two consecutive real nos. are imaginary nos.". I believe these are true vacuously as there don't exist consecutive real nos. Is my understanding correct?

Yes, that is right, as long as what you mean by "consecutive" is clear to me (two real numbers are consecutive if there are no real numbers between them). To be exact and precise, the first implication is : "If x,y are consecutive real numbers then x,y are transcendental" which is vacuously true since the premise can never be satisfied.

Similarly for the second one, @Koro.

Thanks a lot sir. You have helped me understand. :)

@Koro Great!

It's been so great and informative talking to you :) Thanks a lot sir.

@Koro Good to know. I've to have dinner so I'll see you!

Yeah yeah sure :) Happy dinner. :)

Has anybody requested to access the question in my drive

@Eisenstein