Limit involving Greatest Integer and Using Squeeze Theorem | A Tricky Limit Question | Calculus

In this calculus video I will show you a very interesting example of limit of a function involving greatest integer function. To find the limit we use squeeze or sandwich theorem.
The greatest integer function is a function that gives the largest integer which is less than or equal to the number x. This function is denoted by [x] or ⌊x⌋ or ⟦x⟧. We will round off the given number to the nearest integer that is less than or equal to the number itself. For example, [2.4] = 2 and [−2.4] = −2. The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part.
Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉. Both notations are now used in mathematics.

[@williampeters71]

got a little confused on your proof just using example of limit 1//x/ where / / = greatest integer If I plug in a number like 1/2.5 then it would become 1/2 the rule states x-1 < /x/ < x it seems that then in this example 2.5-x = 1.5 < 2 < = 2.5 but 1//x/ would mean 1/1.5 = 2/3 < 1/2 < = 0.4 which is not true because fractions reverse the in equality in general? what did I misunderstand?