Does $\leq$ imply $<$? If I know for two numbers $a$ and $b$ that
$${a &lt; b }$$
Then is it correct to say that
$$ a \leq b $$
I know that the second statement is true as long as the first one is.
However is it true that
$$a \leq b  \implies a&lt;b$$
That is does less than or equal imply less than?
$1\leq 1$ (since $1=1$) but does not hold that $1&lt;1$.
You're correct in the first case.
But now consider the proposition $a\leq b.$ In this case if $a=b=4$, $a\leq b$ is true.
However given $a&lt; b$, it is false that $4&lt;4$.  So it is false that $a\leq b \to a\lt b$ is always true.