In $C[0, 1]$, show that $\{f_n\}$, where $f_n(x) = \frac{nx}{n+x}, x \in [0,1]$, is a Cauchy sequence.
I know the definition that a sequence is Cauchy sequence, if there exists a non negative number $m_{°}$ such that $d(x_{n},x_{m})<\epsilon$ for all $n,m\ge m_{°}$.
What kind of elements am I supposed to choose from the sequence to show whether the sequence is Cauchy or not?
I will assume that we are using the supremum metric.
Let $\epsilon>0$ be given. Choose $N \in \mathbb{N}$ such that $\frac{1}{N}<\epsilon$.
Then note that for all $t \in [0,1]$ and $n>m>N$, $$|f_n(t)-f_m(t)|=|\frac{nt}{n+t}-\frac{mt}{m+t}|=t^2|\frac{n-m}{(n+t)(m+t)}|\leq \frac{n-m}{nm}< \frac{1}{m}<\frac{1}{N}<\epsilon$$ Taking supremum on both sides we get that $$d(f_n,f_m)<\epsilon$$ As we wanted to show.