The series of reciprocals of the least common multiples of integers $X_1
Let $X_1, X_2,\dots$ be a sequence of strictly increasing positive integers. 
For each $n\ge1$, let $W_n$ be the least common multiple of the first $n$ terms $X_1, X_2,\dots, X_n$ of the sequence.
I need to prove the following statement:

The series $1/W_1+1/W_2+\dots+1/W_n\;(n\to\infty)$ is a
  convergent series.

I tried several ways, including the hint given below, but I have no luck to overcome this problem. I'd like to learn the methods that can be used to prove that such a series converges. 

$\newcommand{\lcm}{\operatorname{lcm}}$
Note that for $n>1$ the least common multiple $W_n$ of natural numbers $1\leq X_1<X_2<\ldots<X_n$ fulfills
\begin{align*}
  \lcm(X_{n-1},X_n) \leq W_n
  \end{align*}

The following theorem was originally conjectured by P. Erdös:
Theorem: Let $X_0,\ldots,X_n$ be integers satifying $1\leq X_0<X_1<\ldots<X_n$ then
\begin{align*}
\frac{1}{\lcm(X_0,X_1)}+\frac{1}{\lcm(X_1,X_2)}+\cdots+\frac{1}{\lcm(X_{n-1},X_n)}\leq 1-\frac{1}{2^n}\tag{1}
\end{align*}
From (1) we obtain by noting that $W_1=\lcm(1,X_1)$ 
\begin{align*}
  \frac{1}{W_1}&+\frac{1}{W_2}+\frac{1}{W_3}+\cdots+\frac{1}{W_n}\\
&\leq \frac{1}{W_1}+\frac{1}{\lcm(X_1,X_2)}+\frac{1}{\lcm(X_2,X_3)}+\cdots+\frac{1}{\lcm(X_{n-1},X_n)}\\
&\leq 1-\frac{1}{2^{n}}
  \end{align*}
and OPs claim follows.

A proof of the theorem was given by D. Borwein in the paper A sum of reciprocals of least common multiples.

Proof (D. Borwein): For $j=1,2,\ldots,n$ let $$S_j=\frac{1}{\lcm(X_0,X_1)}+\cdots+\frac{1}{\lcm(X_{j-1},X_j)}$$
then $\lcm(X_{j-1},X_j)=u_jX_{j-1}=v_jX_{j}$ where $u_j>v_j\geq 1$. Hence
  \begin{align*}
  \frac{1}{\lcm(X_{j-1},X_j)}\leq \frac{1}{X_j}\tag{2}
  \end{align*}
  It follows from (2) that
  \begin{align*}
  \frac{1}{\lcm(X_{j-1},X_j)}\leq(u_j-v_j)\frac{1}{\lcm(X_{j-1},X_j)}=\frac{1}{X_{j-1}}-\frac{1}{X_j}\tag{3}
  \end{align*}
  We obtain from (3)
  \begin{align*}
  S_j\leq\frac{1}{X_0}-\frac{1}{X_j}\tag{4}
  \end{align*}
Now we show (1) is valid by considering all possible conditions on $X_0,X_1,\ldots,X_n$
Case 1: $X_n\leq 2^n$. It follows from (4),
\begin{align*}
S_n\leq 1 - \frac{1}{X_n}\leq 1-\frac{1}{2^n}
\end{align*}
Case 2: $X_j>2^j$ for $1\leq j \leq n$. We obtain by (2)
  \begin{align*}
  S_n\leq\frac{1}{X_1}+\frac{1}{X_2}+\cdots+\frac{1}{X_n}<\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}=1-\frac{1}{2^n}
  \end{align*}
Case 3: $X_k\leq2^k$ for some positive integer $k<n$ and $X_j>2^j$ for $k+1\leq j\leq n$. Then, by (2) and (4)
  \begin{align*}
  S_n&=S_k+\frac{1}{\lcm(X_{k},X_{k+1})}+\cdots+\frac{1}{\lcm(X_{n-1},X_n)}\\
&<1-\frac{1}{2^k}+\frac{1}{2^{k+1}}+\cdots+\frac{1}{2^n}\\
&=1-\frac{1}{2^n}
  \end{align*}
We conclude (1) holds in all cases and the theorem follows.

