limit of convergence of continuous functions

Question

Let $f_n,f:X \to \mathbb{R}$ be continuous functions such that $\frac{1}{n}f_{n}(x) \to f(x).$ Is it true that $|\frac{1}{n-K}f_n(x)-\frac{n-H}{n}f(x)| \to 0$ as $n \to \infty$ for some number $H, K \in \mathbb{N}?$

Attempt: I think that is true as $\frac{n-H}{n} \to 1$ when $n \to \infty.$

Answer 1

$$\frac{1}{n-K}f_n(x)= \frac{n}{n-K} \frac{1}{n}f_n(x) \to f(x)$$

and

$$\frac{n-H}{n}f(x) \to f(x).$$

This gives

$$|\frac{1}{n-K}f_n(x)-\frac{n-H}{n}f(x)| \to 0.$$