Convergent Continuous Maps are Homotopic

Question

Let $(f_n)$ be a sequence of continuous functions converging to $f:\mathbb{R}^k\rightarrow \mathbb{R}^l$ in the topology of compact convergence. In this post the answer says that eventually we can related the two by the (sequence of) homotopies $$ F_n(x,t) = tf_n(x) + (1-t)f(x). $$ But why can we do this evenutally?

Answer 1

All maps $f,g : X \to \mathbb R^l$ are homotopic via the straight-line homotopy $H(x,t) = tf(x) + (1-t)g(x)$.