every real number has exactly one integer part I am self studying book Analysis I by Tao, there is an exercise on proving: 
Exercise 5.4.3:
for every real number x, there is exactly one integer N such that $$N \leq x \lt N+1$$
Can anyone give some hints for me to continue my thinking on the proof?
 A: Suppose that there's at least two integers such that the property holds, and suppose that $M<N$.
Now, $M<N \le x <M+1<N+1$ according to the property, but we have that $M<N<M+1$ , but clearly because they're integers (is there an integer $N$ such that there is integers $M$ and $M+1$ and $M<N<M+1$ holds) , $M=N$ should also hold and we have a contradiction.
A: Let there be $N_1,N_2,\dots N_n$ such that $N_n\le x < N_n+1$, $n\in \{1,2,3,\dots \}$
For simplicity let us that $N_1$ and $N_2$ for a while  and then we will generalize for all $n\in \{1,2,3,\dots \}$.
So let us assume that  $N_1\le x < N_1+1$ and $N_2\le x < N_2+1$
Case I
Let $x$ be an integer 
then $N_1=x$ and $N_2=x \implies N_1=x=N_2 \implies N_1=N_2$ 
$\therefore$ for all $n\in \{1,2,3,\dots \}$ $N_1=N_2=N_3=\dots =N_n$
Therefore we have unique integer.
Case II
Let $x$ be a non-integer real number.
Since we have $N_1\le x < N_1+1$
therefore we can write $x=N_1+0.def\dots$ where $d,e,f$ are individual digits.
Similarly we have $x=N_2+0.abx\dots$ where $a,b,c$ are individual digits.
Without loss of generality we can assume that $N_1 > N_2$ 
Therefore $N_1=N_2+m$ where $m\in \mathbb{N}$ 
$\therefore x=N_1+0.def\dots = N_2+m+0.xyz\dots=N_2+m.xyz\dots$
$\therefore x>N_2+1$ and so we have a contradiction.
Therefore we have unique $N$ that satisfies the inequality.
Case-by-case analysis completes our proof.
Note

"$\dots $" in $"0.abc\dots $" means that there may be any number of digits after the decimal point.

A: Uniqueness is easy.
To prove existence, use the Archimedean property to show that there is some integer $M$ which is smaller than x, and an integer larger than x. Show that the set of integers which are at least $M$ and at most $x$ is finite, and use the maximum value of this set.
