Suppose that $B: H \times K \Rightarrow F$ is a continuous bilinear function, where $H,K$ and $F$ are real normed spaces.
I have to prove (not as homework) that if $B(h,k) = o(\lVert(h,k)\rVert^2)$, then $B=0$.
Since $B$ is bilinear and continuous, we have that $\lVert B(h,k)\rVert \leq \lVert B\rVert \lVert h \rVert \lVert k\rVert \leq \lVert B\rVert \lVert(h,k)\rVert^2$, where $\lVert B\rVert$ is the operator norm.
Hence we have $0=\lim_{(h,k)\rightarrow 0} \frac{\lVert B(h,k)\rVert}{\lVert (h,k) \rVert^2} \leq \lim_{(h,k)\rightarrow 0} \frac{\lVert B\rVert \lVert(h,k)\rVert^2}{\lVert (h,k) \rVert^2}= \lVert B\rVert.$
If there is some way for me to get that the last limit is also $0$, I have what has to be proven. But I don't see a way to this. Could anyone provide me with a tiny hint? (No full answers please)