Last day in physics teacher said that any number divided into infinitely many pieces is zero.It got me thinking in kind of weird direction so here is what I was thinking about and how I tried to disprove that number divided into infinite pieces is 0.
For example take number any irrational number and let it be between 1 and 2.Lets call that number $x$ and let $y=x-1$
Now we can have an injection from set of natural numbers(not including 0) into interval [0,y],now let F be injection such that natural number n is mapped to the real number such that it lies in the given interval,its n-th decimal is equal to n-th decimal of y,and all decimals before it are 0.
For example if y=0.431232426577... then F(3)=0.001
Now we have practically divided the irrational number x into infinitely many pieces since,the infinite sum y+$\sum_{i} F(i)$ approaches x for any partial sum.
Thus we have proven that number divided into infintely many pieces is necesarrily zero.How true is this,am I on a right path? Does it even make sense to talk about dividing into infinitely many pieces?
Any number divided by an infinite number of equal parts is zero
$$ \lim_{x\to\infty} \frac{n}{x}= n\lim_{x\to\infty} \frac{1}{x} =0,\quad \forall n\in\mathbb{R} $$ As mentioned by paw88789, in the comments. Also note that in this case, it is more mathematically precise to use the word by as opposed to the more ambiguous word into.