Approximation implies Homotopy

Question

Let $M$ and $N$ be manifolds .I have recently stumbled upon the fact that if $f_n\rightarrow f$ in the strong topology $C_S^r(M,N)$, then we will have that there exists a subsequence of maps that are homotopic to $f$. After looking things up all the proofs that I found use tubular neighborhoods, things that I have yet not seen . Is there another way to prove this statement without using tubular neighborhoods ? Thanks in advance.