# Sequence of functions: Cauchy or not

The question is

$f_n(x) = ax^n + b \cos(x/n)$ is a sequence of functions where $f_n: [0,1] \to \mathbb{R}$. Determine for which $a,b \in \mathbb{R}$ values, $f_n$ is Cauchy w.r.t. the sup-norm in $C[0,1]$.

Since all $a,b,x$ and $n$ are variables, I can not imagine how the graph would be like for different values. I know I should show sufficient effort but in these type of questions I usually use graph of the function thus I really do not know where to start.

Can you give me some hints?

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You must not think of this as a function of 4 variables, but as functions of $x$. You are supposed to look at fixed pairs $a, b$ and decide for such pairs whether the sequence with index $n$ converges uniformly or not. – user20266 Dec 16 '11 at 11:55

Some hints: $C[0,1]$ is complete; so, $\{f_n\}$ is Cauchy if and only if it converges to some $f\in C[0,1]$ (so, this $f$ must be continuous). Also, if $\{f_n\}$ converges in $C[0,1]$, it must converge to its pointwise limit.
I want to check something. The pointwise limit of $f_n$ is a+b when $x=1$ and b otherwise. Then it converges to a continuous function only if $a=0$. You said "the sequence is not Cauchy except in one special case", but isn't it Cauchy only if a=0? – marvinthemartian Dec 16 '11 at 12:59