Why does $(n+1)^n\lt n^{n+1} \implies \left(1+\frac{1}{n}\right)^n\lt n$?

Question

During an example done in lecture, I encountered an inequality by the form of $$(n+1)^n\lt n^{n+1}$$ My professor immediately simplified it to $$\left(1+\frac{1}{n}\right)^n\lt n$$ I have attempted to figure out how this simplification works, for example by taking the $(n+1)^\text{th}$ root of both sides, but nothing seem to lead me anywhere. How is this simplification done?

Answer 1

Notice that: $$ (n + 1)^n = (n(1 + \tfrac{1}{n}))^n = n^n(1 + \tfrac{1}{n})^n $$ so all we have to do is divide both sides by $n^n > 0$.