Exercise 6, § 3

Let $\mathbf{f}$ be a continuous vector function on an interval $[x_0, +\infty[$ of $\mathbf{R}$ and such that for all $\lambda \geqslant 0$ the function $\mathbf{f}(x + \lambda) - \mathbf{f}(x)$ tends to 0 when $x$ tends to $+\infty$.

a) Show that $\mathbf{f}(x + \lambda) - \mathbf{f}(x)$ tends uniformly to 0 with $1/x$ when $\lambda$ belongs to any compact interval $K = [a, b]$ of $[0, +\infty[$. (Argue by contradiction: if there exists a sequence $(x_n)$ tending to $+\infty$ and a sequence of points $(\lambda_{n_k})$ of $K$ such that

for every $n$, then there exists a neighbourhood $J_n$ of $\lambda_n$ in $K$ such that, for every $\lambda \in J_n$, one has $\|\mathbf{f}(x_n + \lambda) - \mathbf{f}(x_n)\| > \alpha$. Construct by induction a decreasing sequence of closed intervals $I_k \subset K$ and a subsequence $(x_{n_k})$ of $(x_n)$ such that $\|\mathbf{f}(x_{n_k} + \lambda) - \mathbf{f}(x_{n_k})\| \geqslant \alpha/3$ for every $\lambda \in I_k$; remark for this that if $\delta_k$ is the length of $I_k$ and $q$ an integer such that $q \delta_k > b - a$, then $\|\mathbf{f}(x + \delta_k) - \mathbf{f}(x)\| \leqslant \alpha/3q$ once $x$ is sufficiently large.)

b) Deduce from a) that $\int_x^{x+1} \mathbf{f}(t)\,dt - \mathbf{f}(x)$ tends to 0 as $x$ tends to $+\infty$, and conclude that $\mathbf{f}(x) = o(x)$.