5.5 التاريخ الموضوع : H.w/ let S is Sampl Let G = {A: A countibu Spance. or A° countibu} Show that whether 6 is 6-felid or net?

7. Consider the set Seq[ℕ] of all the infinite sequences (a_k)_{k ∈ ℕ} = (a_0, a_1, a_2, a_3, ...) of natural numbers. A sequence (a_k)_{k ∈ ℕ} is constant if there is some number n ∈ ℕ so that a_k = n, for all k ∈ ℕ. (a) Construct an injection i: ℕ → Seq[ℕ]. (b) Prove that there is no surjection f: ℕ → Seq[ℕ]. (Consider the arguments of the proof of Cantor’s Theorem that we saw in class.) (c) Deduce from above that ℵ₀ < |Seq[ℕ]|. (Actually, it can be proven that |Seq[ℕ]| = |ℝ|.)

Use a combinatorial proof to show that n(n − 1) - n - -2 (k-2)= = = k(k − 1) n