Let $\left\{u_i\right\}_{i=1}^m$ be a sequence of real vectors, i.e. $u_i\in R^n, 1\le i\le m $ such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. Define:

$$K_m:=\left(\sum_{i=1}^m u_i u_i^T\right)^{-1}u_m$$

Show that $\left\| K_m\right\| $ tends to zero as $m\longrightarrow \infty$.

If $u_i$s are scalar, i.e. $n=1$, the proof seems trivial (not that trivial, but easy to get). I have not been able to find a counter-example for the cases where $u_i$s are vectors (and believe me, I have tried hard). So I think this is very likely to be true in general.

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. Define:

$$K_m:=\left(\sum_{i=1}^m u_i u_i^T\right)^{-1}u_m$$

Show that $\left\| K_m\right\| $ tends to zero as $m\longrightarrow \infty$.

If $u_i$s are scalar, i.e. $n=1$, the proof seems trivial (not that trivial, but easy to get). I have not been able to find a counter-example for the cases where $u_i$s are vectors (and believe me, I have tried hard). So I think this is very likely to be true in general.