enum State
{
    Skip,
    Num
}

void Main()
{
    CheckNumbers("Testing").Dump("Should be true");
    CheckNumbers("11 day of sunshine, 22 days of rain, 34 days of gray sky, then the days number 45").Dump("Should be true");
    CheckNumbers("11 day of sunshine, 22 days of rain, 34 days of gray sky, then the days number 33").Dump("Should be false");
    CheckNumbers("11 day of sunshine, 22 days of rain, 11 days of gray sky, then the days number 45").Dump("Should be false");
}

bool CheckNumbers(string input)
{
    var curState = State.Skip;
    var value = 0;
    var prevValue = -1;

    foreach (var ch in input)
    {
        switch (curState)
        {
            case State.Skip:
                if (char.IsDigit(ch))
                {
                    value = ch - '0';
                    curState = State.Num;
                }
                break;

            case State.Num:
                if (char.IsDigit(ch))
                {
                    value = value * 10 + ch - '0';
                }
                else
                {
                    if (value <= prevValue)
                    {
                        return false;
                    }
                    curState = State.Skip;
                    prevValue = value;
                }
                break;
        }
    }
    return prevValue == -1 || curState == State.Skip || value > prevValue;
}