how do i solve this Algorithm, proof by induction?

let the array be: a

and the array is an array of integers n+1 such that.

a[1] <= a[0] and a[n - 1] <= a[n]

how do i deduce that a must contain a local minimum in
a[1]....a[n-1].

(a[k] is a local minimum if it satisfies)
a[k] <= a[k - 1] and a[k] <= a[k + 1]

i'm not sure but do i have to do it by proof by induction.
can anyone help please?

This is a regular math problem, not an algorithm

If a[1] is not a local minimum, it is a "decreasing point" (what does this mean?). Since the array is finite, there is a largest k such that a[k] is a decreasing point. k < n-1 (why?). What are the possibilities for a[k+1]?

That's an outline of one possible proof. I left it incomplete; it's best for you to fill the gaps yourself. I hope that helps.

many thanks!

You can do it in this way ..let the array be in increasing
order(In calculas we say function is increasing)
=>a[1]>a[0],a[2]>a[1].............a[n]>a[n-1]
=>our assumption is wrong...
Take the second case
let the array be in decreasing order
=>a[1]<a[0],a[2]<a[1].......a[n]<a[n-1]
=>our assumption is wrong...
So the array is decreasing in some range and increasing in some range .
In this case it is decreasing initially and then increasing
So there is a local minima...
The problem can be converted for local maxima also

Such proofs are called as prooving by contradiction