sequences and series - Convergence of $u_n$ defined by $sumlimits_{k=1}^{u_n-1}frac{1}kleq n

For every positive integer $n$, define the positive integer $u_n$ by the condition that
$$\sum\limits_{k=1}^{u_n-1}\frac{1}k\leq n<\sum\limits_{k=1}^{u_n}\frac{1}k$$
Let $w_n=\frac1{u_n}$. Does the sequence $(w_n)$ converge and how to prove rigorously it does?

What I did:

I proved that $u_n$ exists and is unique, and that $u_1=2$ and $u_2=4$.

Proof of uniqueness:

Suppose that $u_n>v_n$ satisfying the defining condition, then $v_n\leq u_n-1$ so $\sum\limits_{k=1}^{v_n}\frac{1}k\leq n$, which is absurd, thus $u_n$ = $v_n$.

Answer

$$n\lt\sum_{k=1}^{u_n}\frac1k\leqslant1+\int_1^{u_n}\frac{\mathrm dt}t=1+\log u_n\implies\frac1{u_n}\lt\frac{\mathrm e}{\mathrm e^n}$$