Is the Sequence an+2=an+1+an Monotonically Increasing?

[analysis001]

Homework Statement

Prove that a_n+2=a_n+1+a_n where a₁=1 and a₂=1 is monotonically increasing.

Homework Equations

A sequence is monotonically increasing if a_n+1≥a_n for all n$\in$N.

The Attempt at a Solution

Base cases:
a₁≤a₂ because 1=1.
a₂≤a₃ because 1<2.

Am I supposed to prove that a_n≤a_n+1 now? I'm not sure how to do that.

 

[SammyS]

Homework Statement

Prove that a_n+2=a_n+1+a_n where a₁=1 and a₂=2 is monotonically increasing.

Homework Equations

A sequence is monotonically increasing if a_n+1≥a_n for all n$\in$N.

The Attempt at a Solution

Base cases:
a₁≤a₂ because 1=1.
a₂≤a₃ because 1<2.

Am I supposed to prove that a_n≤a_n+1 now? I'm not sure how to do that.

a₂ ≥ a₁ because 2 ≥ 1 . After all, a₂ = 2 and a₁ = 1 .

Now, what you need to do:

Assume that the statement is true for some k, where k ≥ 1 .

I.e.:
Assume that a_k+1 ≥ a_k .​

From this, show that it follows that the statement is true for k+1.

I.e.:
Show that a_k+2 ≥ a_k+1 .​