  Let $\eta_{1},\eta_2,...$ be independent (not necessarily identically
distributed) zero-mean random variables (r.v.'s) such that $|\eta_i|\le1$
almost surely for all $i$, and let $Z$ stand for a standard normal r.v. Let
$a_1,a_2,...$ be any real numbers such that $a_1^2+a_2^2+...=1.$ It is shown
that then $$ \P(a_1\eta_1+a_2\eta_2+...\ge x) \le \P(Z\ge x-\la/x) \forall x>0,
$$ where $\la := \ln\frac{2e^3}9=1.495...$. The proof relies on (i) another
probability inequality and (ii) a l'Hospital-type rule for monotonicity, both
developed elsewhere. A multidimensional analogue of this result is given, based
on a dimensionality reduction device, also developed elsewhere. In addition,
extensions to (super)martingales are indicated.
