Is the Medieval proof of the divergence of the Harmonic series valid? As far as I understand it, the Medieval argument for the divergence of the Harmonic series:
$S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\ldots$
relies on replacing each term with the largest power of $2$ less than or equal to it, term by term, giving:
$S'=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\ldots$
and concluding, that since the $i^{\text{th}}$ element of $S'\leq$ the $i^{\text{th}}$ element of $S$ for all $i$, $S'\leq S$. But $S'=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\ldots$, which diverges, and therefore so does $S$.
My question is, to what extent is it legitimate to argue like this, replacing each term in an infinite series with a smaller term and drawing a valid conclusion from that?
For instance if I were to argue that I had the series:
$T=2+3+4+5+\ldots$
And I then subtracted, term by term, $1$ from each element of $T$, giving:
$T'=1+2+3+4+\ldots$
Then the $i^{\text{th}}$ element of $T'\leq$ the $i^{\text{th}}$ element of $T$, as before, suggesting that $T'\leq T$, according to this argument. Except in this case you can see that $T'=T+1>T$. So arguing like this seems to lead to an absurdity.
So my question is, to what extent is element-by-element subtraction like in the Medieval Harmonic series argument legitimate? What is the difference between the Harmonic series argument and my (absurd) $T$ argument?
 A: You are almost there. If $T$ was a real number, you’d be able to conclude that $T+1<T,$ which wouldn’t make sense.
So what does that tell you about $T?$

Both proofs use the theorem:

Given two sequences, $a_n,b_n$ with $0\leq b_n\leq a_n$ for all $n,$ then if $\sum_{n=1}^{\infty} b_n$ does not converges to a real number, then neither does $\sum_{n=1}^\infty a_n.$

A: Yes, that method is a correct, it is called the comparison test for series, and the proof is literally a matter of unwinding definitions. For your second question, you're right that the comparison test alone tells you $T'\leq T$. However, both series diverge to infinity (this is a concept which needs to be defined carefully), so in fact $T'=T=\infty$ (again, these equal signs have a specific meaning in terms of limits).
Your confusion arises from the fact that you're used to arithmetic in the real numbers $\Bbb{R}$, which consists of addition of two numbers, and multiplication of two numbers, and the trichotomy of order relations (i.e $=$ or $>$ or $<$) between two numbers. Now, if you can add/multiply two things, you can do the same for finitely many (this is essentially the priniciple of induction). BUT, it does not mean that ALL the things you're familiar with about arithmetic of real numbers carries over to limits and in particular to the concept of infinity; after all that's why it's a non-trivial subject and why everything needs a careful definition. So, once we define everything carefully, one would say something like "$T'=T+1=\infty$".
