Sqrt(x) function implementation; what is $(i^2 \leq x) \land ((i + 1)^2 > x)$ checking?

Question

Recently I was working on a LeetCode question

Compute and return the square root of x, where x is guaranteed to be a non-negative integer.

Since the return type is an integer, the decimal digits are truncated and only 
the integer part of the result is returned.

Example 1:

Input: 4
Output: 2
Example 2:

Input: 8
Output: 2
Explanation: The square root of 8 is 2.82842..., and since 
             the decimal part is truncated, 2 is returned.

Here is a solution that I really like and understand, except for one line. It is using binary search to get to the solution:

int mySqrt(int x) {
    if (x == 0) return x;
    int left = 1, right = INT_MAX, mid;
    while(left < right) {
        mid = left + (right - left)/2;
        if (mid <= x / mid && (mid + 1) > x / (mid + 1)) {
            return mid; 
        }
        if (mid < x/mid) {
            left = mid + 1;
        } else { // mid > x/mid
            right = mid;
        }
    }
    return right;
}

The conceptual question is: why is it true that given a particular number, say $i$, $$(i^2 \leq x) \land ((i + 1)^2 > x)$$ returns whether or not $i$ is the truncated integer representation of the square root of $x$? (The code block above returns on the identical condition, but inequality is rearranged to avoid integer overflow)

Answer 1

I'll put the two inequalities together, for readability: $i^2 \leq x < (i+1)^2$.

Taking the positive square of a positive number is a monotone function. Hence, $$i^2 \leq x < (i+1)^2 \implies i \leq |\sqrt{x}| < i +1$$

Then, the interger part of $|\sqrt{x}|$ is obviously $i$.