Given a book with $100$ pages and a $100$ lemmas, prove that there is some lemma written on the same page as its index 
A book consists of 100 pages and contains 100 lemmas and some images. Each lemma is at most one page long and can't be split into two pages (it has to fit in one page). The lemmas are numbered from 1 to 100 and are written in ascending order. Prove that there must be at least one lemma written on a page with the same number as the lemma's number.

If lemma no. $1$ is written on page $1$, then it is proved. Let's assume lemma no. $1$ is written on page  $k, k>1$. Then on at least one page there must be $2$ lemmas. Let's assume that always on page $k+i$ we have lemma no. $i+1$ and so on. Then the last $100-k-i$ lemmas must fit on the last page, which means that there will be at least one lemma (number $100$) on page $100$.
But I don't know how to express it in a more mathematical way!
Any help?

We claim more generally that a book of $n$ pages and $n$ lemmas numbered $1$ through $n$ has at least one lemma on a page matching its number.
Proof by induction on $n$:  The case $n=1$ is obvious.  Now suppose the statement is true for some $n$, and suppose we have a book of $n+1$ lemmas and $n+1$ pages.  If lemma $n+1$ is on a page numbered less than $n+1$, then lemmas $1$ through $n$ must be on pages $1$ through $n$, and there must be at least one lemma on a same-numbered page by the inductive hypothesis.  If not, then lemma $n+1$ is on page $n+1$, and we're done.
