# Proof of inequality \$\lceil x \rceil \le x+1\$

I went through the Master Theorum extension for floors and ceiling section 4.6.2 in the book Introduction to Algorithms

Using the inequality $$\lceil x \rceil \le x+1$$

But I haven’t seen the inequality anywhere and could not understand the verifiability of inequality.

Instead the Chapter Floors and ceilings defined floors and ceilings as:

$$x-1 \lt \lfloor x \rfloor \le x \le \lceil x \rceil \lt x+1$$

Please clear my doubt over this.

On how to use this identity and which identity to be considered when because both of them define completely different inequalities.

Thank you.