real analysis - Show $1-1/x

Show $1-1/x <\ln(x) for all $x>1$?

My attempt:

(1) $\ln (x)

Suppose $h: \mathbb{R}^+ \to \mathbb{R}, t\mapsto t-1-\ln(t)$, then $h'(t)=0$ if $t=1$. Furthermore $h'(t)>0$ for all $t\in \mathbb{R}^+$.

(2) $1-1/x<\ln(x)$

Suppose $f: \mathbb{R}^+\to \mathbb{R}, t\mapsto \ln(t)-1+\frac{1}{t}$, then $f'(t)=0$ for $t=1$ and $f'(t)>0$ for all $t\in \mathbb{R}^+$.

Therefore the inequality is true.