Hello khorI'm not quite sure what you mean by "... is a proof." The second sentence in the first paragraph (and the third sentence in the second paragraph) is astatement, followed by anexplanationof why that statement is true.

The first one is:

The second one is:Statement

- The range of is

Explanation

- The square of any real number is non-negative (in other words the range can't contain any negative numbers)

- Every non-negative number is the square of its own square root (in other words, every non-negative number is included in the range because there will always exist a corresponding element in the domain, namely, its square root)
I hope that is what you wanted.Statement

- The range of ... is the set of all non-zero real numbers.

Explanation

- (in other words, the reciprocal function is its own inverse. So, since the domain of is all non-zero real numbers the range will also be all non-zero real numbers)

Grandad