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Which of the following sequences are cauchy?

$1.$ $$f_n(x) = \begin{cases} 0 & \text{if } x \notin [n-1,n+1] \\ x-n+1 & \text{if } x\in [n-1,n]\\ x+n-1 & \text{if } x\in [n,n+1] \end{cases}$$ in the space $\left\{f:\mathbb{R}\rightarrow\mathbb{R}\mid f\text{ is continuous and } \int_{+\infty}^{-\infty}|f(t)|\,dt<\infty\right\}$

$2.$ sequence $f_n(x)=\frac{x+n} n$ in the space $C[0,1]$ with sup-norm metric.

$3.$ sequence $f_n(x)=\frac{nx}{1+nx} $ in the space $C[0,1]$ with sup-norm metric.

I just know that the sequence $\langle \frac{x+n} n \rangle$ is uniformly convergent to $1$ i.e. $\sup|\frac{x+n} n - 1|\rightarrow0$ so is a cauchy sequence. In part $3$ sequence is not uniformly convergent but i don't know what is exact method for cauchy sequence. How to handle part $1$ and part $3?$ Please help me. Thanks.

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    For the first example, I suggest you actually draw the function $|f_n-f_m|$, and find it's integral. I do not think it changes with the value of $n$ and $m$, hence the first sequence is not Cauchy, right? As you said, the second sequence is uniform. – 2017-01-07

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