Suppose we have a regular pencil battery which supplies DC voltage $V$. Say we take copper wire and connect the ends of the battery to an $R$ ohms resistance.

Then Ohm's law tells use the current in the wire is $ \frac{V}{R}$.

This means as we keep decreasing the value of $R$, we will keep getting higher and higher values of current, since $V$ is fixed.

Now if we simply connect the ends of the battery by a copper wire without an intermediate resistance, of course the value of the current will not be infinity, but it will be $\frac{V}{R_{copper}}$ which is still very large.

The resistance of copper is so small that even for $V=1.5$ volts we will get current of larger than 1 Amp with copper. And according to this link 1 Amp can almost give you a heart attack.

So why is it that we dont hear about major accidents about people connecting two ends of a pencil battery with regular copper wire? Is there some fallacy in my reasoning above?