Function from R^2 to R

[o.m.]

Let f be a continous function from \( R^2 \rightarrow R \) and

\( (b_{n}),\ (c_{n}) \) real sequences such that

\( f(b_{n}) \) tend to \( \infty \) when n tend to \( \infty \)

and

\( f(c_{n}) \) tend to \( -\infty \) when n tend to \( \infty \).

Prove that there exists \( (a_{n}) \) a real sequence such that \( f(a_{n})=0 \) for any \( n\in N \) and \( ||a_{n}|| \) tend to \( \infty \) when n tend to \( \infty \).

(Here ||.|| is the euclidean norm in \( R^2 \), \( ||(x,y)||=sqrt{x^2 +y^2} \))