If the signs at the two ends of an interval are different, like f(1) = −1 and f(2) = 2, the graph must pass through 0 somewhere in between — if you draw the curve without lifting your hand, it has no choice but to pass through it. This is the Intermediate Value Theorem. Use bisection to repeatedly narrow the interval and actually find the location of √2.
Intermediate Value Theorem: If a function f(x) is continuous on the closed interval [a,b] and f(a) and f(b) have different values, then for any value k between f(a) and f(b), there must be a c in the interval (a,b) such that f(c)=k. In particular, if f(a) and f(b) have different signs (one positive and one negative), there must be a c such that f(c)=0.
Bisection Method is a practical way to use this theorem. Check the function value at the midpoint m of the interval, keep only the half where the sign changes, and repeat. The interval containing the root gets narrower and narrower. By continuing, you can narrow down the root to the desired precision.
Extreme Value Theorem uses a similar idea — a function that is continuous on a closed interval must attain both a maximum and a minimum somewhere on that interval (the graph must have no breaks and the interval must be closed for this to hold).