Lebesgue measure on countable union

Question

Given the Borel set $A=\cup_{n=1}^{\infty}[n,n+1/n]$ how do I find its Lebesgue measure?

My attempt: Given $\lambda$ we have that \begin{align*} \lambda(\cup_{n=1}^{\infty}[n,n+1/n])&=\lambda([1,2]\cup[2,2+1/4])+\sum_{n=3}^{\infty}\lambda([n,n+1/n]) &&\text{Additive}\\ &=\lambda([1,2+1/2]) +\sum_{n=3}^{\infty}\frac{1}{n}&&\\ &=1+1/2+\sum_{n=3}^{\infty}\frac{1}{n}&&\\ &=\sum_{n=1}^{\infty}\frac{1}{n} \end{align*}

The harmonic series diverges, hence $\lambda(A)=+\infty$.

Is this true? Any help is appreciated.

Answer 1

It seems to me that you miscalculated the measure, although the mistake is not essential in terms of the result, which is simply $\sum\limits_{n=1}^\infty \frac{1}{n} = \infty$.