In the following question,
Let $f_n$ be a sequence of continuous functions defined in R. Suppose that $|f_{n+1}(x)−
f_n(x)| ≤ \frac {n^2}{1+2^n}$
for all x ∈ R and for all n ≥ 1. Show that $f_n$ converges uniformly to some continuous
function $f(x)$ defined in R.
I was thinking of using induction, since I know that $|f_{n+1}(x)−
f_n(x)| ≤ \frac {n^2}{1+2^n}$, then from $f_n$ to $f_{n+1}$ it would be uniformly convergent and $f_n$ is a sequence of continuous functions, then there exists some $\delta > 0$ such that for all $\varepsilon >0$, ${\displaystyle |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .}$ but I'm not too sure how to format it in order to make it clear
Best Answer
Hint:
$$|f_n(x)-f_m(x)|$$ $$ = |f_n(x)-f_{n-1}(x)+f_{n-1}-f_{n-2} +\cdots f_{m+1}(x)-f_m(x)|$$ $$ \le |f_n(x)-f_{n-1}(x|+|f_{n-1}-f_{n-2}| +\cdots |f_{m+1}(x)-f_m(x)|.$$