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

It had the following statement:

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.