Lebesgue integral of $-\frac{1}{n}$

Question

I have following sequence:

$$ f_n = -\frac{1}{n} $$

I wanted to show, that it converges to $f = 0$, but my book says, that it doesn't, because the condition

$$ \int_{\mathbb R} f_1 d\lambda > -\infty $$

is not met.

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My question is - How can I compute Lebesgue integral of $f_1$ ?

$$ f_1 = - \frac{1}{1} = -1 $$

And the integral is:

$$ \int_{\mathbb R} -1 d\lambda $$

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I know, that the integral is defined like this:

$$ \int_E f d\lambda = sup \left\{ \int_E s d\lambda: 0 \leq s \leq f \right\}, $$

where $s$ is considered a simple function, so its integral is defined like this:

$$ \int_E s d\lambda = \sum^k_{i=1} \alpha_i \lambda(A_i \cap E), $$

where $A_i = \{x \in \mathbb R^n: s(x) = \alpha_i \in \mathbb R^+_0\}$.

But still, unfortunately, I'm not able to "connect" all those definitions and compute the integral...