In my real analysis text, an example of proof by induction is given by proving that for any real $x\ge 0$ and all integers $n\ge 0$ $$(1+x)^n \ge 1+nx+\frac {n(n-1)}2x^2$$
I can follow and understand the proof but then the next line after the proof is
It follows that $$(1+x)^n\ge 1+nx$$ and $$(1+x)^n \gt \frac{n(n-1)}{2}x^2$$ hold for every nonnegative ...
How do these stricter results follow from the above? Wouldn't I need to do a proof by induction for each of these also?
Once we have established $$(1+x)^n \ge 1 + nx + \frac{n(n-1)}{2}x^2$$ we also have $$1 + nx + \frac{n(n-1)}{2}x^2 > \frac{n(n-1)}{2}x^2$$ and $$1 + nx + \frac{n(n-1)}{2}x^2 \ge 1+nx$$ (Where we have strict inequality when $x > 0$). Thus the weaker inequalities follow: $$(1+x)^n \ge 1 + nx$$ $$(1+x)^n \gt \frac{n(n-1)}{2}x^2$$ with strict inequality when $x > 0$.