I'm looking for examples of facts/results in basic school that are supported by calculus/analysis. Here is an example:
Consider the infinite decimal expansion (a usual topic in basic school): basic school means primary;secondary school but mainly high school!
$a = 0.12345678910\ 11\ 12\ 13\ 14\ldots$ (the Champernowne constant)
What does this expression mean? The answer $a$ is the limit of the sequence:
$a_1 = 0.1, a_2 = 0.12, a_3 = 0.123,$ etc.
How do we know that this sequence has a limit? Here calculus helps: every bounded monotonic sequence has limit.
Another example: Why does there exist a number $a$ such that $a \cdot a = 11$?
Answer from calculus: $f(x) = x^2$ is continuous, $f(0) = 0$ and $f(4) = 16$ as $0 \leq 11 \leq 16$, by the Intermediate value theorem the existence of the number $a$ (square root of $11$) follows.