Show that if $a < b$ and $a \leq c$ then $b \leq c$

Question

Let $a,b,c \in \mathbb{R}$.

Show that if $a < b$ and $a \leq c$ then $b \leq c$.

I think it's false because when $a=c$ then $b>c$, which contradicts $b \leq c$. Yet, it's true according to the solution to a bigger problem I have. Either the solution is wrong or I'm wrong at something so basic.

Answer 1

This is a false statement. Consider $a= 1$, $b = 3$, $c=2$.

Answer 2

$1<3$ and $1\leq2$, but $3>2.$