Prove certain set is convex I'm having trouble proving this statement:

Prove that the set
$$X=\{ (x,y)\in\mathbb{R}^2 : y\geq 0, x\leq b+y \}$$
given $b\in\mathbb{R}$ is convex.

The work I've done so far:
I first assumed we havo two different points $(x_1,y_1),(x_2,y_2)\in X$. Then, the segment between them is made of the points
$$(\lambda x_1 + (1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2) \ , \ \lambda\in[0,1].$$
So if I prove that points are inside $X$ for any $\lambda$ I would have proven the convexity.
It's easy to see that
$$\lambda y_1 + (1-\lambda)y_2 \geq 0,$$
since it's a positive number plus another positive number. Now, I don't know how to prove that
$$\lambda x_1 + (1-\lambda)x_2 \leq b + \lambda y_1 + (1-\lambda)y_2$$
knowing obviously that $x_1 \leq b+y_1$ and $x_2\leq b+y_2$.
What can I do to prove this part? Any help or hint will be appreciated, thanks in advance.

Let (x₁,y₁),(x₂,y₂) ∈ X
Let (c,d) = (λx₁ + (1-λ)x₂, λy₁ + (1-λ)y₂)
c = λx₁ + (1-λ)x₂ ≤ λ(b + y₁) + (1-λ)(b + y₂)
= (λb + b - λb) + λy₁ + (1-λ)y₂
= b + d
hence c ≤ b + d
and you have already proved why d ≥ 0
It's actually pretty straight forward. You should've just expanded the expression.
