An infinity of natural numbers for which a function doesn't have a limit at $\infty$

Question

Let $f,g:[0,\infty)\rightarrow \mathbb{R}$ be two continuous functions with the property that $\lim_{x \to \infty} f(x)=\lim_{x \to \infty} g(x)=\infty$. Prove that there is an infinity of natural numbers $k$ for which $h_k:[0,\infty)\rightarrow \mathbb{R},h_k(x)=f(k+\{g(x)\})$ doesn't have a limit as $x \to \infty$.($\{a\}$ denotes the fractional part of the real number $a$)
My attempt : $k+\{g(x)\}=\lfloor{k}\rfloor$,so $\lim_{x \to \infty}h_k(x)=f(\lim_{x \to \infty}\lfloor{k}\rfloor)=\infty$ and this is not what I need to prove.Why is my approach wrong and how should I solve this problem?

Answer 1

Your argument isn't correct, and we need additional assumptions to deduce the desired conclusion, as is pointed out on the above comments. So, I'll presume that $f,g$ are continuous. Fix $k$ and suppose $h_k$ tends to $L_k$ as $x\to\infty$. For any given $M>0$, since $\lim_{x\to\infty}g(x) = \infty$, we can find $x>M$ such that $g(x)>g(M)+3$. Then there exists a natural number $N$ such that $g(M)\le N<N+1\le g(x)$. By intermediate value theorem, $[N,N+1]\subset g([M,\infty))$ for all $M>0$. This implies $[0,1)\subset \{g\}([M,\infty))$ for all $M>0$. Now, it follows that for any $u\in [0,1)$, we can find a sequence $x_n \to \infty$ such that $\{g(x_n)\} = u$ for all $n$. This implies $$ L_k=\lim_{x\to\infty}h_k(x) = \lim_{n\to\infty} h_k(x_n) = f(k+u) $$ for all $u\in [0,1)$. That is, $f(x)= L_k$ on $x\in [k,k+1)$.

Now, assume to the contrary that $\lim_{x\to\infty}h_k(x)$ does not exist for at most a finite number of $k$'s. Then, there is $k_0$ such that $$ L_k=\lim_{x\to\infty}h_k(x) $$ exists for all $k\ge k_0$. This means $f(x)=L_k$ on $[k,k+1)$ as we've seen in the above argument. If $f(x) = L_{k+1} $ on $[k+1,k+2)$, then by continuity of $f$, it must be $L_k = L_{k+1}$. This readily shows that $L_{k_0} = L_k$ for all $k\ge k_0$. This says that $f(x) = L_{k_0}$ for all $x\ge k_0$, and leads to a contradiction that $\lim_{x\to\infty} f(x) = \infty$.