Multiply infinity by a positive number

[PirateFan308]

Homework Statement

Prove that if limX_n = +∞ and limY_n>0 then limX_nY_n=+∞

The Attempt at a Solution

limX_nY_n = limX_nlimY_n = (c)(+∞) where c is a positive real number

I know in my head that a positive number multiplied by infinity is positive, but I am unsure how to prove this and we have not yet done this particular example in class.

 

[Deveno]

instead of thinking of ∞ as a number, think of:

$$\lim_{n \to \infty} x_n = \infty$$

meaning, no matter how large a positive real number N we choose, for all large enough n, x_n > N.

now, suppose

$$\lim_{n \to \infty} y_n = L > 0$$

for large enough n, we can ensure that y_n > L/2 > 0.

can we make x_ny_n larger than any positive real number M?

(what happens if we pick n so that x_nis larger than 2M/L, and y_n is larger than _____?)