(A) (3 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be a continuous function such that $f ( r ) = f \left( r + \frac { 1 } { n } \right)$ for each $r \in \mathbb { Q }$ and each positive integer $n$. Prove or disprove the following statement: $f$ is a constant function. (B) (7 marks) Let $a _ { n } , n \geq 1$ be a sequence of non-negative real numbers such that $a _ { m + n } \leq a _ { m } + a _ { n }$ for all $m , n$. Show that $$\lim _ { n \longrightarrow \infty } \frac { a _ { n } } { n } = \inf \left\{ \left. \frac { a _ { n } } { n } \right\rvert\, n \geq 1 \right\}$$
(A) (3 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be a continuous function such that $f ( r ) = f \left( r + \frac { 1 } { n } \right)$ for each $r \in \mathbb { Q }$ and each positive integer $n$. Prove or disprove the following statement: $f$ is a constant function.\\
(B) (7 marks) Let $a _ { n } , n \geq 1$ be a sequence of non-negative real numbers such that $a _ { m + n } \leq a _ { m } + a _ { n }$ for all $m , n$. Show that
$$\lim _ { n \longrightarrow \infty } \frac { a _ { n } } { n } = \inf \left\{ \left. \frac { a _ { n } } { n } \right\rvert\, n \geq 1 \right\}$$